## Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force . Part A What is the ratio ? ANSWER: Correct Conceptual Question 13.3 A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth. Part A What is the ratio of the force on the first satellite to that on the second satellite? ANSWER: Correct F1 F2 F1 F2 = 2 F1 F2 kg kg F1 F2 = 0.682 F1 F2 Part B What is the ratio of the acceleration of the first satellite to that of the second satellite? ANSWER: Correct Problem 13.2 The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: Correct Part B What is the ratio of this gravitational force to the weight of the 90.0 ball? ANSWER: a1 a2 = 1 a1 a2 kg g cm 1.11×10−8 N g 1.26×10−8 Correct Problem 13.6 The space shuttle orbits 310 above the surface of the earth. Part A What is the gravitational force on a 7.5 sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth. Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. km kg Fe on s = 67.0 N kg km Fgrav G N m2/kg2 me kg re m Hint 2. Law of gravitation According to Newton’s law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: ANSWER: Correct Part B What fraction is this of the satellite’s weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: F = Gm1m2/r2 G m1 m2 r r = 7.03×10r 6 m = 1.86×10Fgrav 4 N g m/s2 0.824 Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth’s gravity is simply the square of the ratio of the earth’s radius to the sum of the earth’s radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut’s weight is never zero. When people speak of “weightlessness” in space, what they really mean is “free fall.” Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: Correct w = G m/ me r2e Fgrav = Gmem/(re + h)2 w [re/(re + h)]2 gmoon = 1.62 m s2 gJupiter = 25.9 m s2 Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth’s surface at a speed of 1.90×104 . You may want to review ( pages 362 – 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket’s initial kinetic energy in terms of its unknown mass, ? What is the rocket’s initial gravitational potential energy in terms of its unknown mass, ? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket’s kinetic energy when it is very far away from the Earth? Therefore, what is the rocket’s velocity when it is very far away from the Earth? ANSWER: Correct Problem 13.13 Part A m/s m m 1.54×104 ms What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid’s orbital radius? Express your answer with the appropriate units. ANSWER: Correct Part B What is the asteroid’s orbital speed? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s = 3.89×1011 R m = 1.85×104 v ms Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon’s orbit are 2 . 1.01×107 m 4920 ms km/s Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: Correct Part B The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 . Part A What is the period of the oscillations? Express your answer with the appropriate units. v1 = 11.3 km s km v2 = cm cm s ANSWER: Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part D What is the amplitude? Express your answer with the appropriate units. 2.91 s 0.344 Hz 2.16 rad s ANSWER: Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object’s displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the 28.0 cm 60.5 cms F F = −kx x k m k x = 0 block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we’ve assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds. The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). A A x = A remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right. x = −A x = −A x = −A x = −A x = A T f f = 1/T f Hz Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct unchanged. doubled. halved. s s f f = 10 Hz Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Hz T T = 0.025 s A Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: Correct Now assume for the remaining Parts G – J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q T T K and L K and M K and P L and N M and P m s T t = 0 a aT = 0.005 s a period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? T T = 0.02 s t t = 0.01 s d d = 0.48 m d Express your answer in meters. ANSWER: Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct d = 0.36 m A = 20.0 cm Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: f = 0.25 Hz 0 = -60 % s t = 0 s cm cm/s 0 = -2.09 rad Correct Part B What is the phase at ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: t = 0.5 s = 0 rad t = 1.0 s = 2.09 rad t = 1.5 s = 4.19 rad Correct Problem 14.12 A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: Correct Problem 14.14 The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct g cm s 3.00 Nm x(t) = (2.0 cm)cos(10t − /4) t 2.00 cm Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 0.628 s 5.00 Nm -0.785 rad Part E The initial coordinate of the mass. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part F The initial velocity. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part G The maximum speed. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 1.41 cm 14.1 cms 20.0 cms Part H The total energy. Express your answer to one decimal place and include the appropriate units. ANSWER: Correct Part I The velocity at . Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 14.17 A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds. You may want to review ( pages 389 – 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? 1.0 mJ t = 0.40 s 1.46 cms N/m cm s Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? ANSWER: Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: Correct Changing the Period of a Pendulum m = 110 g vmax = 49 cms A simple pendulum consisting of a bob of mass attached to a string of length swings with a period . Part A If the bob’s mass is doubled, approximately what will the pendulum’s new period be? Hint 1. Period of a simple pendulum The period of a simple pendulum of length is given by , where is the acceleration due to gravity. ANSWER: Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. m L T T L T = 2 Lg −− g T/2 T ‘2T 2T g/6 ANSWER: Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth’s gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: Correct In the space station, where all objects undergo the same acceleration due to the earth’s gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. T/6 T/’6 ‘6T 6T It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. Problem 14.20 A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil. g % s L = 19 cm m lmoon = 0.35 m m g 1.0 MHz Part A The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won’t rupture the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the disk’s maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 94.2%. You received 135.71 out of a possible total of 144 points. N amax = 6.6 μm vmax = 41 ms

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## CE 309 Fluid Mechanics Laboratory 2015 Assignment: ABET Criterion b You are tasked by SMU to design laboratory equipment for accurately determining discharge coefficients of an orifice in a reservoir discharging into the atmosphere (free jet). The equipment will be used in an undergraduate fluid mechanics laboratory class. You are not allowed to recommend an over-the-shelf system sold by manufacturers but must begin with basic materials. Your design must include the following; • Neat sketches and drawing illustrating your design. Sketches must be to scale. All sections of the sketch must be labeled in detail. As an example, a proposed motor must show the type, horsepower as well as any details necessary for the acquisition of the motor. • Statement of cost of individual items as well as the gross. It must also include installation costs where applicable. You are encouraged to recommend modern instrumentation in you design however costs must be kept as reasonable as possible. An esoteric system with no regard to the cost is of little value. Justify all your choices. • Develop a procedure for students operating the system to achieve the laboratory objectives. Indicate the advantages of your design over the current. • Keep your report to 3 pages maximum.

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## Chapter 12 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Spinning Grinding Wheel At time a grinding wheel has an angular velocity of 26.0 . It has a constant angular acceleration of 33.0 until a circuit breaker trips at time = 1.80 . From then on, the wheel turns through an angle of 432 as it coasts to a stop at constant angular deceleration. Part A Through what total angle did the wheel turn between and the time it stopped? Express your answer in radians. You did not open hints for this part. ANSWER: Part B At what time does the wheel stop? Express your answer in seconds. You did not open hints for this part. ANSWER: t = 0 rad/s rad/s2 t s rad t = 0 rad Part C What was the wheel’s angular acceleration as it slowed down? Express your answer in radians per second per second. You did not open hints for this part. ANSWER: An Exhausted Bicyclist An exhausted bicyclist pedals somewhat erratically when exercising on a static bicycle. The angular velocity of the wheels follows the equation , where represents time (measured in seconds), = 0.500 , = 0.250 and = 2.00 . Part A There is a spot of paint on the front wheel of the bicycle. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the wheel’s rotation. What angular displacement has the spot of paint undergone between time 0 and 2 seconds? Express your answer in radians using three significant figures. s rad/s2 (t) = at − bsin(ct) for t 0 t a rad/s2 b rad/s c rad/s t = 0 = 0 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Express the angular displacement undergone by the spot of paint at seconds in degrees. Remember to use the unrounded value from Part A, should you need it. Express your answer in degrees using three significant figures. You did not open hints for this part. ANSWER: Part C What distance has the spot of paint moved in 2 seconds if the radius of the wheel is 50 centimeters? Express your answer in centimeters using three significant figures. You did not open hints for this part. ANSWER: = rad t = 2 = d Typesetting math: 29% Part D Which one of the following statements describes the motion of the spot of paint at seconds? You did not open hints for this part. ANSWER: Flywheel Kinematics A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration . The flywheel is assumed to be at rest at time in Parts A and B of this problem. Part A Find the time it takes to accelerate the flywheel to if the angular acceleration is . Express your answer in terms of and . d = cm t = 2.0 The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is increasing. t = 0 t1 1 1 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Find the angle through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocity . Express your answer in terms of some or all of the following: , , and . You did not open hints for this part. ANSWER: Part C Assume that the motor has accelerated the wheel up to an angular velocity with angular acceleration in time . At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of . Find , the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of some or all of the following: , \texttip{\alpha }{alpha}, and \texttip{t_{\rm 1}}{t_1}. You did not open hints for this part. t1 = 1 1 1 t1 1 = 1 t1 −5 t2 1 Typesetting math: 29% ANSWER: Surprising Exploding Firework A mortar fires a shell of mass \texttip{m}{m} at speed \texttip{v_{\rm 0}}{v_0}. The shell explodes at the top of its trajectory (shown by a star in the figure) as designed. However, rather than creating a shower of colored flares, it breaks into just two pieces, a smaller piece of mass \large{\frac15m} and a larger piece of mass \large{\frac45m}. Both pieces land at exactly the same time. The smaller piece lands perilously close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance \texttip{d}{d} from the mortar. If there had been no explosion, the shell would have landed a distance \texttip{r}{r} from the mortar. Assume that air resistance and the mass of the shell’s explosive charge are negligible. Part A Find the distance \texttip{d}{d} from the mortar at which the larger piece of the shell lands. Express \texttip{d}{d} in terms of \texttip{r}{r}. You did not open hints for this part. \texttip{t_{\rm 2}}{t_2} = s Typesetting math: 29% ANSWER: Kinetic Energy of a Dumbbell This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy \texttip{K_{\rm total}}{K_total} of a dumbbell of mass \texttip{m}{m} when it is rotating with angular speed \texttip{\omega }{omega} and its center of mass is moving translationally with speed \texttip{v}{v}. Denote the dumbbell’s moment of inertia about its center of mass by \texttip{I_{\rm cm}}{I_cm}. Note that if you approximate the spheres as point masses of mass m/2 each located a distance \texttip{r}{r} from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by I_{\rm cm} = mr^2, but this fact will not be necessary for this problem. Part A Find the total kinetic energy \texttip{K_{\rm tot}}{K_tot} of the dumbbell. Express your answer in terms of \texttip{m}{m}, \texttip{v}{v}, \texttip{I_{\rm cm}}{I_cm}, and \texttip{\omega }{omega}. You did not open hints for this part. \texttip{d}{d} = Typesetting math: 29% ANSWER: Part B This question will be shown after you complete previous question(s). Unwinding Cylinder A cylinder with moment of inertia \texttip{I}{I} about its center of mass, mass \texttip{m}{m}, and radius \texttip{r}{r} has a string wrapped around it which is tied to the ceiling . The cylinder’s vertical position as a function of time is y(t). At time t = 0 the cylinder is released from rest at a height \texttip{h}{h} above the ground. Part A The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate \texttip{\omega }{omega} and \texttip{v}{v}, the velocity of the center of mass of the cylinder? \texttip{K_{\rm tot}}{K_tot} = Typesetting math: 29% Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity. Express \texttip{\omega }{omega} in terms of \texttip{v}{v} and other given quantities. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C Suppose that at a certain instant the velocity of the cylinder is \texttip{v}{v}. What is its total kinetic energy, \texttip{K_{\rm total}}{K_total}, at that instant? Express \texttip{K_{\rm total}}{K_total} in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{I}{I}, and \texttip{v}{v}. You did not open hints for this part. ANSWER: Part D \texttip{\omega }{omega} = \texttip{K_{\rm total}}{K_total} = Typesetting math: 29% Find \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, the cylinder’s vertical velocity when it hits the ground. Express \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, in terms of \texttip{g}{g}, \texttip{h}{h}, \texttip{I}{I}, \texttip{m}{m}, and \texttip{r}{r}. You did not open hints for this part. ANSWER: Kinetic Energy of a Rotating Wheel A simple wheel has the form of a solid cylinder of radius \texttip{r}{r} with a mass \texttip{m}{m} uniformly distributed throughout its volume. The wheel is pivoted on a stationary axle through the axis of the cylinder and rotates about the axle at a constant angular speed. The wheel rotates \texttip{n}{n} full revolutions in a time interval \texttip{t}{t}. Part A What is the kinetic energy \texttip{K}{K} of the rotating wheel? Express your answer in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{n}{n}, \texttip{t}{t} and, \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Finding Torque \texttip{v_{\rm f \hspace{1 pt}}}{v_f} = \texttip{K}{K} = Typesetting math: 29% A force \texttip{\vec{F}}{F_vec} of magnitude \texttip{F}{F} making an angle \texttip{\theta }{theta} with the x axis is applied to a particle located along axis of rotation A, at Cartesian coordinates (0, 0) in the figure. The vector \texttip{\vec{F}}{F_vec} lies in the xy plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the xy plane. A particle is located at a vector position \texttip{\vec{r}}{r_vec} with respect to an axis of rotation (thus \texttip{\vec{r}}{r_vec} points from the axis to the point at which the particle is located). The magnitude of the torque \texttip{\tau }{tau} about this axis due to a force \texttip{\vec{F}}{F_vec} acting on the particle is given by \tau = r F \sin(\alpha) = ({\rm moment \; arm}) \cdot F = rF_{\perp}, where \texttip{\alpha }{alpha} is the angle between \texttip{\vec{r}}{r_vec} and \texttip{\vec{F}}{F_vec}, \texttip{r}{r} is the magnitude of \texttip{\vec{r}}{r_vec}, \texttip{F}{F} is the magnitude of \texttip{\vec{F}}{F_vec}, the component of \texttip{\vec{r}}{r_vec} that is perpendicualr to \texttip{\vec{F}}{F_vec} is the moment arm, and \texttip{F_{\rm \perp}}{F_\perp} is the component of the force that is perpendicular to \texttip{\vec{r}}{r_vec}. Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation. In this problem, you must express the angle \texttip{\alpha }{alpha} in the above equation in terms of \texttip{\theta }{theta}, \texttip{\phi }{phi}, and/or \texttip{\pi }{pi} when entering your answers. Keep in mind that \pi = 180\;\rm degrees and (\pi/2) = 90\;\rm degrees . Part A What is the torque \texttip{\tau_{\rm A}}{tau_A} about axis A due to the force \texttip{\vec{F}}{F_vec}? Express the torque about axis A at Cartesian coordinates (0, 0). You did not open hints for this part. Typesetting math: 29% ANSWER: Part B What is the torque \texttip{\tau_{\rm B}}{tau_B} about axis B due to the force \texttip{\vec{F}}{F_vec}? (B is the point at Cartesian coordinates (0, b), located a distance \texttip{b}{b} from the origin along the y axis.) Express the torque about axis B in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: Part C What is the torque \texttip{\tau_{\rm C}}{tau_C} about axis C due to \texttip{\vec{F}}{F_vec}? (C is the point at Cartesian coordinates (c, 0), a distance \texttip{c}{c} along the x axis.) Express the torque about axis C in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: \texttip{\tau_{\rm A}}{tau_A} = \texttip{\tau_{\rm B}}{tau_B} = Typesetting math: 29% Part D What is the torque \texttip{\tau_{\rm D}}{tau_D} about axis D due to \texttip{\vec{F}}{F_vec}? (D is the point located at a distance \texttip{d}{d} from the origin and making an angle \texttip{\phi }{phi} with the x axis.) Express the torque about axis D in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. ANSWER: Torque Magnitude Ranking Task The wrench in the figure has six forces of equal magnitude acting on it. \texttip{\tau_{\rm C}}{tau_C} = \texttip{\tau_{\rm D}}{tau_D} = Typesetting math: 29% Part A Rank these forces (A through F) on the basis of the magnitude of the torque they apply to the wrench, measured about an axis centered on the bolt. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: The Parallel-Axis Theorem Typesetting math: 29% Learning Goal: To understand the parallel-axis theorem and its applications To solve many problems about rotational motion, it is important to know the moment of inertia of each object involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and tedious process. While it is important to be able to calculate moments of inertia from the definition (I=\sum m_ir_i^2), in most cases it is useful simply to recall the moment of inertia of a particular type of object. The moments of inertia of frequently occurring shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere) are well known and readily available from any mechanics text, including your textbook. However, one must take into account that an object has not one but an infinite number of moments of inertia. One of the distinctions between the moment of inertia and mass (the latter being the measure of tranlsational inertia) is that the moment of inertia of a body depends on the axis of rotation. The moments of inertia that you can find in the textbooks are usually calculated with respect to an axis passing through the center of mass of the object. However, in many problems the axis of rotation does not pass through the center of mass. Does that mean that one has to go through the lengthy process of finding the moment of inertia from scratch? It turns out that in many cases, calculating the moment of inertia can be done rather easily if one uses the parallel-axis theorem. Mathematically, it can be expressed as I=I_{\rm cm}+md^2, where \texttip{I_{\rm cm}}{I_cm} is the moment of inertia about an axis passing through the center of mass, \texttip{m}{m} is the total mass of the object, and \texttip{I}{I} is the moment of inertia about another axis, parallel to the one for which \texttip{I_{\rm cm}}{I_cm} is calculated and located a distance \texttip{d}{d} from the center of mass. In this problem you will show that the theorem does indeed work for at least one object: a dumbbell of length \texttip{2r}{2r} made of two small spheres of mass \texttip{m}{m} each connected by a light rod (see the figure). NOTE: Unless otherwise noted, all axes considered are perpendicular to the plane of the page. Part A Using the definition of moment of inertia, calculate I_{\rm cm}, the moment of inertia about the center of mass, for this object. Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Using the definition of moment of inertia, calculate I_{\rm B}, the moment of inertia about an axis through point B, for this object. Point B coincides with (the center of) one of the spheres (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part C Now calculate I_{\rm B} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: I_{\rm cm} = I_{\rm B} = I_{\rm B} = Typesetting math: 29% Part D Using the definition of moment of inertia, calculate I_{\rm C}, the moment of inertia about an axis through point C, for this object. Point C is located a distance \texttip{r}{r} from the center of mass (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part E Now calculate I_{\rm C} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: Consider an irregular object of mass \texttip{m}{m}. Its moment of inertia measured with respect to axis A (parallel to the plane of the page), which passes through the center of mass (see the second diagram), is given by I_{\rm A}=0.64mr^2. Axes B, C, D, and E are parallel to axis A; their separations from axis A are shown in the diagram. In the subsequent questions, the subscript indicates the axis with respect to which the moment of inertia is measured: for instance, I_{\rm C} is the moment of inertia about axis C. I_{\rm C} = I_{\rm C} = Typesetting math: 29% Part F Which moment of inertia is the smallest? ANSWER: Part G Which moment of inertia is the largest? ANSWER: I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} Typesetting math: 29% Part H Which moments of inertia are equal? ANSWER: Part I Which moment of inertia equals 4.64mr^2? ANSWER: Part J Axis X, not shown in the diagram, is parallel to the axes shown. It is known that I_{\rm X}=6mr^2. Which of the following is a possible location for axis X? ANSWER: I_{\rm A} and I_{\rm D} I_{\rm B} and I_{\rm C} I_{\rm C} and I_{\rm E} No two moments of inertia are equal. I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} between axes A and C between axes C and D between axes D and E to the right of axis E Typesetting math: 29% Torque and Angular Acceleration Learning Goal: To understand and apply the formula \tau=I\alpha to rigid objects rotating about a fixed axis. To find the acceleration \texttip{a}{a} of a particle of mass \texttip{m}{m}, we use Newton’s second law: \vec {F}_{\rm net}=m\vec{a}, where \texttip{\vec{F}_{\rm net}}{F_vec_net} is the net force acting on the particle. To find the angular acceleration \texttip{\alpha }{alpha} of a rigid object rotating about a fixed axis, we can use a similar formula: \tau_{\rm net}=I\alpha, where \tau_{\rm net}=\sum \tau is the net torque acting on the object and \texttip{I}{I} is its moment of inertia. In this problem, you will practice applying this formula to several situations involving angular acceleration. In all of these situations, two objects of masses \texttip{m_{\rm 1}}{m_1} and \texttip{m_{\rm 2}}{m_2} are attached to a seesaw. The seesaw is made of a bar that has length \texttip{l}{l} and is pivoted so that it is free to rotate in the vertical plane without friction. You are to find the angular acceleration of the seesaw when it is set in motion from the horizontal position. In all cases, assume that m_1>m_2, and that counterclockwise is considered the positive rotational direction. Part A The seesaw is pivoted in the middle, and the mass of the swing bar is negligible. Find the angular acceleration \texttip{\alpha }{alpha} of the seesaw. Express your answer in terms of some or all of the quantities \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{l}{l}, as well as the acceleration due to gravity \texttip{g}{g}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B In what direction will the seesaw rotate, and what will the sign of the angular acceleration be? ANSWER: Part C This question will be shown after you complete previous question(s). \texttip{\alpha }{alpha} = The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part D In what direction will the seesaw rotate and what will the sign of the angular acceleration be? ANSWER: Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Pivoted Rod with Unequal Masses The figure shows a simple model of a seesaw. These consist of a plank/rod of mass \texttip{m_{\rm r}}{m_r} and length 2x allowed to pivot freely about its center (or central axis), as shown in the diagram. A small sphere of mass \texttip{m_{\rm 1}}{m_1} is attached to the left end of the rod, and a small sphere of mass \texttip{m_{\rm 2}}{m_2} is attached to the right end. The spheres are small enough that they can be considered point particles. The gravitational force acts downward. The magnitude of the acceleration due to gravity is equal to \texttip{g}{g}. The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part A What is the moment of inertia \texttip{I}{I} of this assembly about the axis through which it is pivoted? Express the moment of inertia in terms of \texttip{m_{\rm r}}{m_r}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. Keep in mind that the length of the rod is 2x, not \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Weight and Wheel Consider a bicycle wheel that initially is not rotating. A block of mass \texttip{m}{m} is attached to the wheel and is allowed to fall a distance \texttip{h}{h}. Assume that the wheel has a moment of inertia \texttip{I}{I} about its rotation axis. Part A Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius \texttip{r_{\mit A}}{r_A} \texttip{I}{I} = Typesetting math: 29% . Find \texttip{\omega _{\mit A}}{omega_A}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit A}}{omega_A} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit A}}{r_A}, and \texttip{I}{I}. You did not open hints for this part. ANSWER: Part B Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius \texttip{r_{\mit B}}{r_B} . Find \texttip{\omega _{\mit B}}{omega_B}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit B}}{omega_B} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit B}}{r_B}, and \texttip{I}{I}. \texttip{\omega _{\mit A}}{omega_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C Which of the following describes the relationship between \texttip{\omega _{\mit A}}{omega_A} and \texttip{\omega _{\mit B}}{omega_B}? You did not open hints for this part. ANSWER: A Bar Suspended by Two Vertical Strings A rigid, uniform, horizontal bar of mass \texttip{m_{\rm 1}}{m_1} and length \texttip{L}{L} is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance d < L/2 from the left end of the bar and is connected to the ceiling; string B is attached to \texttip{\omega _{\mit B}}{omega_B} = \omega_A > \omega_B \omega_B > \omega_A \omega_A = \omega_B Typesetting math: 29% the left end of the bar and is connected to the floor. A small block of mass \texttip{m_{\rm 2}}{m_2} is supported against gravity by the bar at a distance \texttip{x}{x} from the left end of the bar, as shown in the figure. Throughout this problem positive torque is that which spins an object counterclockwise. Use \texttip{g}{g} for the magnitude of the acceleration due to gravity. Part A Find \texttip{T_{\mit A}}{T_A}, the tension in string A. Express the tension in string A in terms of \texttip{g}{g}, \texttip{m_{\rm 1}}{m_1}, \texttip{L}{L}, \texttip{d}{d}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B Find \texttip{T_{\mit B}}{T_B}, the magnitude of the tension in string B. Express the magnitude of the tension in string B in terms of \texttip{T_{\mit A}}{T_A}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. \texttip{T_{\mit A}}{T_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). Part D If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal). What is the smallest possible value of \texttip{x}{x} such that the bar remains stable (call it \texttip{x_{\rm critical}}{x_critical})? Express your answer for \texttip{x_{\rm critical}}{x_critical} in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{d}{d}, and \texttip{L}{L}. You did not open hints for this part. ANSWER: Part E \texttip{T_{\mit B}}{T_B} = \texttip{x_{\rm critical}}{x_critical} = Typesetting math: 29% This question will be shown after you complete previous question(s). A Tale of Two Nutcrackers This problem explores the ways that torque can be used in everyday life. Case 1 To crack a nut a force of magnitude \texttip{F_{\rm n}}{F_n} (or greater) must be applied on both sides, as shown in the figure. One can see that a nutcracker only applies this force at the point in which it contacts the nut (at a distance \texttip{d}{d} from the nutcracker pivot). In this problem the nut is placed in a nutcracker and equal forces of magnitude \texttip{F}{F} are applied to each end, directed perpendicular to the handle, at a distance \texttip{D}{D} from the pivot. The frictional forces between the nut and the nutcracker are equal and large enough that the nut doesn’t shoot out of the nutcracker. Part A Find \texttip{F}{F}, the magnitude of the force applied to each side of the nutcracker required to crack the nut. Express the force in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: Typesetting math: 29% Case 2 The nut is now placed in a nutcracker with only one lever, as shown, and again friction keeps the nut from slipping. The top “jaw” (in black) is fixed to a stationary frame so that a person just has to apply a force to the bottom lever. Assume that \texttip{F_{\rm 2}}{F_2} is directed perpendicular to the handle. Part B Find the magnitude of the force \texttip{F_{\rm 2}}{F_2} required to crack the nut. Express your answer in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: \texttip{F}{F} = \texttip{F_{\rm 2}}{F_2} = Typesetting math: 29% Part C This question will be shown after you complete previous question(s). Precarious Lunch A uniform steel beam of length \texttip{L}{L} and mass \texttip{m_{\rm 1}}{m_1} is attached via a hinge to the side of a building. The beam is supported by a steel cable attached to the end of the beam at an angle \texttip{\theta }{theta}, as shown. Through the hinge, the wall exerts an unknown force, \texttip{F}{F}, on the beam. A workman of mass \texttip{m_{\rm 2}}{m_2} sits eating lunch a distance \texttip{d}{d} from the building. Part A Find \texttip{T}{T}, the tension in the cable. Remember to account for all the forces in the problem. Express your answer in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{L}{L}, \texttip{d}{d}, \texttip{\theta }{theta}, and \texttip{g}{g}, the magnitude of the acceleration due to gravity. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Find \texttip{F_{\mit x}}{F_x}, the \texttip{x}{x}-component of the force exerted by the wall on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T} and other given quantities. You did not open hints for this part. ANSWER: Part C Find \texttip{F_{\mit y}}{F_y}, the y-component of force that the wall exerts on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T}, \texttip{\theta }{theta}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. ANSWER: \texttip{T}{T} = \texttip{F_{\mit x}}{F_x} = \texttip{F_{\mit y}}{F_y} = Typesetting math: 29% Pulling Out a Nail A nail is hammered into a board so that it would take a force \texttip{F_{\rm nail}}{F_nail}, applied straight upward on the head of the nail, to pull it out. (Take an upward force to be positive.) A carpenter uses a crowbar to try to pry it out. The length of the handle of the crowbar is \texttip{L_{\rm h}}{L_h}, and the length of the forked portion of the crowbar (which fits around the nail) is \texttip{L_{\rm n}}{L_n}. Assume that the forked portion of the crowbar is perfectly horizontal. The handle of the crowbar makes an angle \texttip{\theta }{theta} with the horizontal, and the carpenter pulls directly along the horizontal. Typesetting math: 29% Part A With what force \texttip{F_{\rm pull}}{F_pull} must the carpenter pull on the crowbar to remove the nail? Express the force in terms of \texttip{F_{\rm nail}}{F_nail}, \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta}. You did not open hints for this part. ANSWER: Now, imagine that \texttip{F_{\rm pull}}{F_pull} is not large enough to dislodge the nail. In other words, the nail stays in place, and, if the surface below the crowbar weren’t present, the crowbar would rotate around the point of contact with the nail. This makes it natural to take the pivot point to be the point where the crowbar is in contact with the nail. (But you are always free to choose the pivot point to be any fixed point, even one some distance from the object.) Part B What is the magnitude of the normal force that the surface exerts on the crowbar, \texttip{F_{\rm bar}}{F_bar}? Express your answer for the normal force in terms of \texttip{F_{\rm pull}}{F_pull}, \texttip{\theta }{theta}, \texttip{L_{\rm n}}{L_n}, and \texttip{L_{\rm h}}{L_h}. Take the upward direction to be positive. You did not open hints for this part. ANSWER: Three bars are shown in the figure. Both bars A and B have \texttip{F_{\rm pull}}{F_pull} acting on them in the horizontal direction. Bar C has \texttip{F_{\rm pull}}{F_pull} = \texttip{F_{\rm bar}}{F_bar} = Typesetting math: 29% \texttip{F_{\rm pull}}{F_pull} strictly perpendicular to the bar. \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta} are the same quantities in each case. Part C Let the magnitude of the torque about the bend in the crowbars be denoted \texttip{\tau _{\mit A}}{tau_A}, \texttip{\tau _{\mit B}}{tau_B} and \texttip{\tau _{\mit C}}{tau_C} for each of the three cases shown. Which of the following is the correct relationship between the magnitude of of the torques? You did not open hints for this part. ANSWER: Tipping Crane \tau_A > \tau_B > \tau_C \tau_A > \tau_B = \tau_C \tau_A = \tau_B = \tau_C \tau_A < \tau_B = \tau_C \tau_A < \tau_B < \tau_C \tau_A = \tau_B > \tau_C \tau_A = \tau_B < \tau_C Typesetting math: 29% Learning Goal: To step through the application of \Sigma \vec{\tau} = 0 to prevent a crane from tipping over. A crane of weight \texttip{W}{W} has a length (wheelbase) \texttip{c}{c}, and its center of mass is midway between the wheels (i.e., the mass of the lifting arm is negligible). The arm extending from the front of the crane has a length \texttip{b}{b} and makes an angle \texttip{\theta }{theta} with the horizontal. The crane contacts the ground only at its front and rear tires. Part A While watching the crane in operation, an observer mentions to you that for a given load there is a maximum angle \texttip{\theta _{\rm max}}{theta_max} between 0 \degree and 90 \degree that the crane arm can make with the horizontal without tipping the crane over. Is this correct? ANSWER: Part B Later that week, while watching the same crane in operation, a different observer mentions to you that there is a maximum load the crane can lift without tipping, and you can find that maximum load by observing the minimum angle \texttip{\theta _{\rm min}}{theta_min} that the crane arm makes with the horizontal. Is this correct? ANSWER: yes no Typesetting math: 29% Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H yes no Typesetting math: 29% Notice that we have the weight of the crane exerting a torque about the front wheels of the same crane. To create a torque, a force must be present, so it would seem that somehow the weight of the crane is exerting a force upon its front wheels. However, the crane is one object, and it follows from Newton's laws that an object cannot exert a net force upon itself. This crane seems to be defying Newton's laws. What's going on here? ANSWER: Part I Assume you get a summer job as a crane operator. On the first day you are lifting a heavy piece of machinery. Even though you have the arm at 70^\circ above the horizontal, the crane begins to tip slowly forward. Consider the following possible actions: Release the brake on the lifting cable so that the load accelerates 1. downward. 2. Release the lifting arm so that \texttip{\theta }{theta} decreases rapidly and the load accelerates downward. 3. Increase \texttip{\theta }{theta} while simultaneously letting out the lifting cable so that the load accelerates downward. 4. Put the crane wheels in gear and accelerate the crane forward. None of these solutions is ideal, but which will have the short-term effect of restoring contact of the crane's rear wheels with the ground? ANSWER: Spinning Situations Suppose you are standing on the center of a merry-go-round that is at rest. You are holding a spinning bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above. Newton's laws don't apply to torques. The rear wheels exert a downward force on the front wheels. The crane is not accelerating so forces don't matter. The earth exerts forces on the crane and the load. all but 1 all but 2 all but 3 all but 4 all of them Typesetting math: 29% For this problem, neglect any air resistance or friction between the merry-go-round and its foundation. Part A Suppose you now grab the edge of the wheel with your hand, stopping it from spinning. What happens to the merry-go-round? You did not open hints for this part. ANSWER: Twirling a Baton A majorette in a parade is performing some acrobatic twirlings of her baton. Assume that the baton is a uniform rod of mass 0.120{\rm kg} and length 80.0{\rm cm} . Part A Initially, the baton is spinning about a line through its center at angular velocity 3.00{\rm rad/s} . What is its angular momentum? Express your answer in kilogram meters squared per second. It remains at rest. It begins to rotate counterclockwise (as observed from above). It begins to rotate clockwise (as observed from above). Typesetting math: 29% You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \rm kg \cdot m^2/s Typesetting math: 29%

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## Essay on A la carte cable versus current cable companies: should consumers be allowed to pick and choose the channels they watch and pay for?

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## Assignment 12 Due: 11:59pm on Friday, May 9, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 15.6 A 2.00 -diameter vat of liquid is 2.90 deep. The pressure at the bottom of the vat is 1.20 . Part A What is the mass of the liquid in the vat? Express your answer with the appropriate units. ANSWER: Correct Problem 15.8 A 120-cm-thick layer of oil floats on a 130-cm-thick layer of water. Part A What is the pressure at the bottom of the water layer? Express your answer with the appropriate units. ANSWER: Correct m m atm 6490 kg p = 1.25×105 Pa Problem 15.9 A research submarine has a 40.0 -diameter window 8.00 thick. The manufacturer says the window can withstand forces up to 1.20×106 . What is the submarine’s maximum safe depth? Part A The pressure inside the submarine is maintained at 1.0 atm. Express your answer with the appropriate units. ANSWER: Correct Problem 15.13 Part A What is the minimum hose diameter of an ideal vacuum cleaner that could lift a 13 dog off the floor? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 15.40 The 80 student in the figure balances a 1300 elephant on a hydraulic lift. cm cm N 947 m kg d = 4.0 cm kg kg Typesetting math: 100% You may want to review ( pages 415 – 419) . For help with math skills, you may want to review: Rearrangement of Equations Involving Multiplication and Division Part A What is the diameter of the piston the student is standing on? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Given that the height of the fluid on the two sides is the same in the figure, how is the pressure of the fluid on the two sides related? What is the definition of pressure? What is the area of the right cylinder? What is the force exerted by the elephant on the right cylinder? What is the additional pressure above atmospheric pressure in the fluid under the elephant? What is the additional pressure above atmospheric pressure under the student in the left cylinder? What is the force exerted by the student on the left cylinder? What is the area of the left cylinder? ANSWER: Correct Part B d = 0.50 m Typesetting math: 100% When a second student joins the first, the piston sinks 30 . What is the second student’s mass? Express your answer to two significant figures and include the appropriate units. You did not open hints for this part. ANSWER: Enhanced EOC: Problem 15.17 A 4.70 rock whose density is 4300 is suspended by a string such that half of the rock’s volume is under water. You may want to review ( pages 419 – 423) . For help with math skills, you may want to review: Conversion Factors Part A What is the tension in the string? Express your answer with the appropriate units. Hint 1. How to approach the problem What are the three forces acting on the rock? Draw a picture indicating the direction of the forces on the rock and an appropriate coordinate system indicating the positive direction. How is volume related to mass and density? What is the volume of the rock? What is the buoyant force on the rock given that half of the rock is underwater? What is the gravitational force on the rock? Given that the rock is suspended, what is the net force on the rock? Now, determine the tension in the string. cm m = kg kg/m3 Typesetting math: 100% ANSWER: Correct Problem 15.15 A block floats in water with its long axis vertical. The length of the block above water is 1.0 . Part A What is the block’s mass density? Express your answer with the appropriate units. ANSWER: Correct Crown of Gold? According to legend, the following challenge led Archimedes to the discovery of his famous principle: Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense) metal had been substituted for some of the gold, then the crown would displace more water than the pure gold. A similar method of answering the challenge, based on the same physical principle, is to compute the ratio of the actual weight of the crown, , and the apparent weight of the crown when it is submerged in water, . See whether you can follow in Archimedes’ footsteps. The figure shows what is meant by weighing the crown while it is submerged in water. 40.7 N 2.0 cm × 2.0 cm × 8.0 cm cm 875 kg m3 Wactual Wapparent Typesetting math: 100% Part A Take the density of the crown to be . What is the ratio of the crown’s apparent weight (in water) to its actual weight ? Express your answer in terms of the density of the crown and the density of water . Hint 1. Find an expression for the actual weight of the crown Assume that the crown has volume . Find the actual weight of the crown. Express in terms of , (the magnitude of the acceleration due to gravity), and . ANSWER: Hint 2. Find an expression for the apparent weight of the crown Assume that the crown has volume , and take the density of water to be . Find the apparent weight of the crown submerged in water. Express your answer in terms of , (the magnitude of the acceleration due to gravity), , and . Hint 1. How to approach the problem c Wapparent Wactual c w V Wactual Wactual V g c Wactual = cV g V w Wapparent V g w c Typesetting math: 100% The apparent weight of the crown when it is submerged in water will be less than its actual weight (weight in air) due to the buoyant force, which opposes gravity. Hint 2. Find an algebraic expression for the buoyant force. Find the magnitude of the buoyant force on the crown when it is completely submerged in water. Express your answer in terms of , , and the gravitational acceleration . ANSWER: ANSWER: ANSWER: Correct Part B Imagine that the apparent weight of the crown in water is , and the actual weight is . Is the crown made of pure (100%) gold? The density of water is grams per cubic centimeter. The density of gold is grams per cubic centimeter. Hint 1. Find the ratio of weights for a crown of pure gold Given the expression you obtained for , what should the ratio of weights be if the crown is made of pure gold? Express your answer numerically, to two decimal places. Fbuoyant w V g Fbuoyant = wV g Wapparent = (c − w)gV = Wapparent Wactual 1 − w c Wapparent = 4.50 N Wactual = 5.00 N w = 1.00 g = 19.32 Wapparent Wactual Typesetting math: 100% ANSWER: ANSWER: Correct For the values given, , whereas for pure gold, . Thus, you can conclude that the the crown is not pure gold but contains some less-dense metal. The goldsmith made sure that the crown’s (true) weight was the same as that of the amount of gold he was allocated, but in so doing was forced to make the volume of the crown slightly larger than it would otherwise have been. Problem 15.23 A 1.0-cm-diameter pipe widens to 2.0 cm, then narrows to 0.5 cm. Liquid flows through the first segment at a speed of 2.0 . Part A What is the speed in the second segment? Express your answer with the appropriate units. ANSWER: Correct = 0.95 Wapparent Wactual Yes No = 4.50/5.00 = 0.90 Wapparent Wactual = 1 − = 0.95 Wapparent Wactual w g m/s 0.500 ms Typesetting math: 100% Part B What is the speed in the third segment? Express your answer with the appropriate units. ANSWER: Correct Part C What is the volume flow rate through the pipe? Express your answer with the appropriate units. ANSWER: Correct Understanding Bernoulli’s Equation Bernoulli’s equation is a simple relation that can give useful insight into the balance among fluid pressure, flow speed, and elevation. It applies exclusively to ideal fluids with steady flow, that is, fluids with a constant density and no internal friction forces, whose flow patterns do not change with time. Despite its limitations, however, Bernoulli’s equation is an essential tool in understanding the behavior of fluids in many practical applications, from plumbing systems to the flight of airplanes. For a fluid element of density that flows along a streamline, Bernoulli’s equation states that , where is the pressure, is the flow speed, is the height, is the acceleration due to gravity, and subscripts 1 and 2 refer to any two points along the streamline. The physical interpretation of Bernoulli’s equation becomes clearer if we rearrange the terms of the equation as follows: . 8.00 ms 1.57×10−4 m3 s p1 +gh1 + = +g + 1 2 v21 p2 h2 1 2 v22 p v h g p1 − p2 = g(h2 −h1)+ ( − ) 1 2 v22 v21 Typesetting math: 100% The term on the left-hand side represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from point 1 to point 2. The two terms on the right-hand side represent, respectively, the change in potential energy, , and the change in kinetic energy, , of the unit volume during its flow from point 1 to point 2. In other words, Bernoulli’s equation states that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the change in potential and kinetic energy per unit volume that occurs during the flow. This is nothing more than the statement of conservation of mechanical energy for an ideal fluid flowing along a streamline. Part A Consider the portion of a flow tube shown in the figure. Point 1 and point 2 are at the same height. An ideal fluid enters the flow tube at point 1 and moves steadily toward point 2. If the cross section of the flow tube at point 1 is greater than that at point 2, what can you say about the pressure at point 2? Hint 1. How to approach the problem Apply Bernoulli’s equation to point 1 and to point 2. Since the points are both at the same height, their elevations cancel out in the equation and you are left with a relation between pressure and flow speeds. Even though the problem does not give direct information on the flow speed along the flow tube, it does tell you that the cross section of the flow tube decreases as the fluid flows toward point 2. Apply the continuity equation to points 1 and 2 and determine whether the flow speed at point 2 is greater than or smaller than the flow speed at point 1. With that information and Bernoulli’s equation, you will be able to determine the pressure at point 2 with respect to the pressure at point 1. Hint 2. Apply Bernoulli’s equation Apply Bernoulli’s equation to point 1 and to point 2 to complete the expression below. Here and are the pressure and flow speed, respectively, and subscripts 1 and 2 refer to point 1 and point 2. Also, use for elevation with the appropriate subscript, and use for the density of the fluid. Express your answer in terms of some or all of the variables , , , , , , and . Hint 1. Flow along a horizontal streamline p1 − p2 g(h2 − h1) 1 ( − ) 2 v22 v21 p v h p1 v1 h1 p2 v2 h2 Typesetting math: 100% Along a horizontal streamline, the change in potential energy of the flowing fluid is zero. In other words, when applying Bernoulli’s equation to any two points of the streamline, and they cancel out. ANSWER: Hint 3. Determine with respect to By applying the continuity equation, determine which of the following is true. Hint 1. The continuity equation The continuity equation expresses conservation of mass for incompressible fluids flowing in a tube. It says that the amount of fluid flowing through a cross section of the tube in a time interval must be the same for all cross sections, or . Therefore, the flow speed must increase when the cross section of the flow tube decreases, and vice versa. ANSWER: ANSWER: h1 = h2 p1 + = 1 2 v21 p2 + v2 2 2 v2 v1 $V A $t $V = = $t A1v1 A2v2 v2 > v1 v2 = v1 v2 < v1 Typesetting math: 100% Correct Thus, by combining the continuity equation and Bernoulli's equation, one can characterize the flow of an ideal fluid.When the cross section of the flow tube decreases, the flow speed increases, and therefore the pressure decreases. In other words, if , then and . Part B As you found out in the previous part, Bernoulli's equation tells us that a fluid element that flows through a flow tube with decreasing cross section moves toward a region of lower pressure. Physically, the pressure drop experienced by the fluid element between points 1 and 2 acts on the fluid element as a net force that causes the fluid to __________. Hint 1. Effects from conservation of mass Recall that, if the cross section of the flow tube varies, the flow speed must change to conserve mass. This means that there is a nonzero net force acting on the fluid that causes the fluid to increase or decrease speed depending on whether the fluid is flowing through a portion of the tube with a smaller or larger cross section. ANSWER: Correct Part C Now assume that point 2 is at height with respect to point 1, as shown in the figure. The ends of the flow tube have the same areas as the ends of the horizontal flow tube shown in Part A. Since the cross section of the flow tube is decreasing, Bernoulli's equation tells us that a fluid element flowing toward point 2 from point 1 moves toward a region of lower pressure. In this case, what is the pressure drop The pressure at point 2 is lower than the pressure at point 1. equal to the pressure at point 1. higher than the pressure at point 1. A2 < A1 v2 > v1 p2 < p1 A v decrease in speed increase in speed remain in equilibrium h Typesetting math: 100% experienced by the fluid element? Hint 1. How to approach the problem Apply Bernoulli's equation to point 1 and to point 2, as you did in Part A. Note that this time you must take into account the difference in elevation between points 1 and 2. Do you need to add this additional term to the other term representing the pressure drop between the two ends of the flow tube or do you subtract it? ANSWER: Correct Part D From a physical point of view, how do you explain the fact that the pressure drop at the ends of the elevated flow tube from Part C is larger than the pressure drop occurring in the similar but purely horizontal flow from Part A? The pressure drop is smaller than the pressure drop occurring in a purely horizontal flow. equal to the pressure drop occurring in a purely horizontal flow. larger than the pressure drop occurring in a purely horizontal flow. Typesetting math: 100% Hint 1. Physical meaning of the pressure drop in a tube As explained in the introduction, the difference in pressure between the ends of a flow tube represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from one end to the other end of the flow tube. ANSWER: Correct In the case of purely horizontal flow, the difference in pressure between the two ends of the flow tube had to balance only the increase in kinetic energy resulting from the acceleration of the fluid. In an elevated flow tube, the difference in pressure must also balance the increase in potential energy of the fluid; therefore a higher pressure is needed for the flow to occur. Venturi Meter with Two Tubes A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe, a configuration called a Venturi meter. Consider such an arrangement with a horizontal pipe carrying fluid of density . The fluid rises to heights and in the two open-ended tubes (see figure). The cross-sectional area of the pipe is at the position of tube 1, and at the position of tube 2. p1 − p2 A greater amount of work is needed to balance the increase in potential energy from the elevation change. decrease in potential energy from the elevation change. larger increase in kinetic energy. larger decrease in kinetic energy. h1 h2 A1 A2 Typesetting math: 100% Part A Find , the gauge pressure at the bottom of tube 1. (Gauge pressure is the pressure in excess of outside atmospheric pressure.) Express your answer in terms of quantities given in the problem introduction and , the magnitude of the acceleration due to gravity. Hint 1. How to approach the problem Use Bernoulli's law to compute the difference in pressure between the top and bottom of tube 1. The pressure at the top of the tube is defined to be atmospheric pressure. Note: Inside the tube, since the fluid is not flowing, the terms involving velocity in Bernoulli's equation can be ignored. Thus, Bernoulli's equation reduces to the formula for pressure as a function of depth in a fluid of uniform density. Hint 2. Simplified Bernoulli's equation For a fluid of uniform density that is not flowing, the pressure at a depth below the surface is given by , where is the pressure at the surface and is the magnitude of the acceleration due to gravity. ANSWER: Correct The fluid is pushed up tube 1 by the pressure of the fluid at the base of the tube, and not by its kinetic energy, since there is no streamline around the sharp edge of the tube. Thus energy is not conserved (there is turbulence at the edge of the tube) at the entrance of the tube. Since Bernoulli's law is essentially a statement of energy conservation, it must be applied separately to the fluid in the tube and the fluid flowing in the main pipe. However, the pressure in the fluid is the same just inside and just outside the tube. Part B Find , the speed of the fluid in the left end of the main pipe. Express your answer in terms of , , , and either and or , which is equal to . p1 g p h p = p0 + gh p0 g p1 = gh1 v1 h1 h2 g A1 A2 A1 A2 Typesetting math: 100% Hint 1. How to approach the problem Energy is conserved along the streamlines in the main flow. This means that Bernoulli's law can be applied to obtain a relationship between the fluid pressure and velocity at the bottom of tube 1, and the fluid pressure and velocity at the bottom of tube 2. Hint 2. Find in terms of What is , the pressure at the bottom of tube 2? Express your answer in terms of , , and any other given quantities. Hint 1. Recall Part A Obtain the solution for this part in the same way that you found an expression for in terms of in Part A of this problem. ANSWER: Hint 3. Find in terms of given quantities Find , the fluid pressure at the bottom of tube 2. Express your answer in terms of , , , , and . Hint 1. Find the pressure at the bottom of tube 2 Find , the fluid pressure at the bottom of tube 2. Express your answer in terms of , , and . ANSWER: p2 h2 p2 h2 g p1 h1 p2 = gh2 p2 p2 p1 v1 A1 A2 p2 p1 v1 v2 p2 = p1 + ( − ) 1 2 v1 2 v2 2 Typesetting math: 100% Hint 2. Find in terms of The fluid is incompressible, so you can use the continuity equation to relate the fluid velocities and in terms of the geometry of the pipe. Find , the fluid velocity at the bottom of tube 2, in terms of . Your answer may include and , the cross-sectional areas of the pipe. ANSWER: ANSWER: ANSWER: Correct Note that this result depends on the difference between the heights of the fluid in the tubes, a quantity that is more easily measured than the heights themselves. Problem 15.39 The container shown in the figure is filled with oil. It is open to the atmosphere on the left. v2 v1 v1 v2 v2 v1 A1 A2 v2 = A1 A2 v1 p2 = p1 + ( )(1 − ) 1 2 v1 2 ( ) A1 A2 2 v1 = 2g h1−h2 ( ) −1 A1 A2 2 −−−−−−−−−−−−−− Typesetting math: 100% Part A What is the pressure at point A? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B What is the pressure difference between points A and B? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct PA = 106 kPa PB − PA = 4.4 kPa Typesetting math: 100% Part C What is the pressure difference between points A and C? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 15.48 You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 32.4 . If you then lower the statue into a tub of water, so that it is completely submerged, the scale reads 17 . Part A What is the density? Express your answer with the appropriate units. ANSWER: Correct Problem 15.60 Water flows from the pipe shown in the figure with a speed of 2.0 . PC − PA = 4.4 kPa N N statue = 2100 kg m3 m/s Typesetting math: 100% Part A What is the water pressure as it exits into the air? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the height of the standing column of water? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again P = 1.0×105 Pa h h = Typesetting math: 100% Score Summary: Your score on this assignment is 83.9%. You received 78.84 out of a possible total of 94 points. Typesetting math: 100%

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## 1) During the late 19th century, the nature of work changed for Americans and has never been the same since. Facets of work which we take for granted today such as working long hours and working by the clock, working with machines, and feeling like a very small part of a very large company or corporation, were alien to workers of the late 19th century. Scholars have long debated this transition in the workplace, and have attempted to assess whether the change was beneficial or not for the worker. Using your own personal experience if you wish, but also using specific historical examples discussed in the text and lesson, do you feel the changes the American worker experienced in the 19th century were beneficial or not? Would you rather work in a pre-industrial workplace, not governed by the clock, or has the advent of machines and machinery allowed American workers more freedom? Or has it made them robots? Also, why do you think we don’t have violent labor conflicts in this country like there were in the 19th century? Are workers happier? Or just used to a system now that they have no choice but to accept?

Planned labor has always been of paramount importance as supporting … Read More...

## Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force . Part A What is the ratio ? ANSWER: Correct Conceptual Question 13.3 A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth. Part A What is the ratio of the force on the first satellite to that on the second satellite? ANSWER: Correct F1 F2 F1 F2 = 2 F1 F2 kg kg F1 F2 = 0.682 F1 F2 Part B What is the ratio of the acceleration of the first satellite to that of the second satellite? ANSWER: Correct Problem 13.2 The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: Correct Part B What is the ratio of this gravitational force to the weight of the 90.0 ball? ANSWER: a1 a2 = 1 a1 a2 kg g cm 1.11×10−8 N g 1.26×10−8 Typesetting math: 100% Correct Problem 13.6 The space shuttle orbits 310 above the surface of the earth. Part A What is the gravitational force on a 7.5 sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth. Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. km kg Fe on s = 67.0 N kg km Fgrav G N m2/kg2 me kg re m Typesetting math: 100% Hint 2. Law of gravitation According to Newton’s law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: ANSWER: Correct Part B What fraction is this of the satellite’s weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: F = Gm1m2/r2 G m1 m2 r r = 7.03×10r 6 m = 1.86×10Fgrav 4 N g m/s2 0.824 Typesetting math: 100% Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth’s gravity is simply the square of the ratio of the earth’s radius to the sum of the earth’s radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut’s weight is never zero. When people speak of “weightlessness” in space, what they really mean is “free fall.” Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: Correct w = G m/ me r2e Fgrav = Gmem/(re + h)2 w [re/(re + h)]2 gmoon = 1.62 m s2 gJupiter = 25.9 m s2 Typesetting math: 100% Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth’s surface at a speed of 1.90×104 . You may want to review ( pages 362 – 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket’s initial kinetic energy in terms of its unknown mass, ? What is the rocket’s initial gravitational potential energy in terms of its unknown mass, ? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket’s kinetic energy when it is very far away from the Earth? Therefore, what is the rocket’s velocity when it is very far away from the Earth? ANSWER: Correct Problem 13.13 Part A m/s m m 1.54×104 ms Typesetting math: 100% What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid’s orbital radius? Express your answer with the appropriate units. ANSWER: Correct Part B What is the asteroid’s orbital speed? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s = 3.89×1011 R m = 1.85×104 v ms Typesetting math: 100% Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon’s orbit are 2 . 1.01×107 m 4920 ms km/s Typesetting math: 100% Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: Correct Part B The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 . Part A What is the period of the oscillations? Express your answer with the appropriate units. v1 = 11.3 km s km v2 = cm cm s Typesetting math: 100% ANSWER: Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part D What is the amplitude? Express your answer with the appropriate units. 2.91 s 0.344 Hz 2.16 rad s Typesetting math: 100% ANSWER: Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object’s displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the 28.0 cm 60.5 cms F F = −kx x k m k x = 0 Typesetting math: 100% block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we’ve assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds. The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). A A x = A remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right. x = −A x = −A x = −A x = −A x = A T f f = 1/T f Hz Typesetting math: 100% Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct unchanged. doubled. halved. s s f f = 10 Hz Typesetting math: 100% Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Hz T T = 0.025 s A Typesetting math: 100% Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: Correct Now assume for the remaining Parts G – J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q T T K and L K and M K and P L and N M and P m s T t = 0 a aT = 0.005 s a Typesetting math: 100% period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? T T = 0.02 s t t = 0.01 s d d = 0.48 m d Typesetting math: 100% Express your answer in meters. ANSWER: Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct d = 0.36 m A = 20.0 cm Typesetting math: 100% Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: f = 0.25 Hz 0 = s t = 0 s cm cm/s Typesetting math: 100% Incorrect; Try Again Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Problem 14.12 A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: 0 = g cm s Typesetting math: 100% Correct Problem 14.14 The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C 3.00 Nm x(t) = (2.0 cm)cos(10t − /4) t 2.00 cm 0.628 s Typesetting math: 100% The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G Typesetting math: 100% This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Enhanced EOC: Problem 14.17 A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds. You may want to review ( pages 389 – 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? N/m cm s Typesetting math: 100% ANSWER: Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: Correct Changing the Period of a Pendulum A simple pendulum consisting of a bob of mass attached to a string of length swings with a period . Part A If the bob’s mass is doubled, approximately what will the pendulum’s new period be? Hint 1. Period of a simple pendulum The period of a simple pendulum of length is given by m = 110 g vmax = 49 cms m L T Typesetting math: 10T0% L , where is the acceleration due to gravity. ANSWER: Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. ANSWER: T = 2 Lg −− g T/2 T &2T 2T g/6 T/6 T/&6 &6T 6T Typesetting math: 100% Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth’s gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: Correct In the space station, where all objects undergo the same acceleration due to the earth’s gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. Problem 14.20 A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. g ( s Typesetting math: 100% ANSWER: Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil. Part A The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won’t rupture the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: L = 19 cm m lmoon = 0.35 m m g 1.0 MHz N amax = 6.6 μm Typesetting math: 100% Correct Part B What is the disk’s maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 81.4%. You received 117.25 out of a possible total of 144 points. vmax = 41 ms

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## Read this article and answer this question in 2 pages : Answers should be from the below article only. What is the difference between “standards-based” and “standards-embedded” curriculum? what are the curricular implications of this difference? Article: In 2007, at the dawn of 21st century in education, it is impossible to talk about teaching, curriculum, schools, or education without discussing standards . standards-based v. standards-embedded curriculum We are in an age of accountability where our success as educators is determined by individual and group mastery of specific standards dem- onstrated by standardized test per- formance. Even before No Child Left Behind (NCLB), standards and measures were used to determine if schools and students were success- ful (McClure, 2005). But, NCLB has increased the pace, intensity, and high stakes of this trend. Gifted and talented students and their teach- ers are significantly impacted by these local or state proficiency stan- dards and grade-level assessments (VanTassel-Baska & Stambaugh, 2006). This article explores how to use these standards in the develop- ment of high-quality curriculum for gifted students. NCLB, High-Stakes State Testing, and Standards- Based Instruction There are a few potentially positive outcomes of this evolution to public accountability. All stakeholders have had to ask themselves, “Are students learning? If so, what are they learning and how do we know?” In cases where we have been allowed to thoughtfully evaluate curriculum and instruction, we have also asked, “What’s worth learning?” “When’s the best time to learn it?” and “Who needs to learn it?” Even though state achievement tests are only a single measure, citizens are now offered a yardstick, albeit a nar- row one, for comparing communities, schools, and in some cases, teachers. Some testing reports allow teachers to identify for parents what their chil- dren can do and what they can not do. Testing also has focused attention on the not-so-new observations that pov- erty, discrimination and prejudices, and language proficiency impacts learning. With enough ceiling (e.g., above-grade-level assessments), even gifted students’ actual achievement and readiness levels can be identi- fied and provide a starting point for appropriately differentiated instruc- tion (Tomlinson, 2001). Unfortunately, as a veteran teacher for more than three decades and as a teacher-educator, my recent observa- tions of and conversations with class- room and gifted teachers have usually revealed negative outcomes. For gifted children, their actual achievement level is often unrecognized by teachers because both the tests and the reporting of the results rarely reach above the student’s grade-level placement. Assessments also focus on a huge number of state stan- dards for a given school year that cre- ate “overload” (Tomlinson & McTighe, 2006) and have a devastating impact on the development and implementation of rich and relevant curriculum and instruction. In too many scenarios, I see teachers teach- ing directly to the test. And, in the worst cases, some teachers actually teach The Test. In those cases, The Test itself becomes the curriculum. Consistently I hear, “Oh, I used to teach a great unit on ________ but I can’t do it any- more because I have to teach the standards.” Or, “I have to teach my favorite units in April and May after testing.” If the outcomes can’t be boiled down to simple “I can . . .” state- ments that can be posted on a school’s walls, then teachers seem to omit poten- tially meaningful learning opportunities from the school year. In many cases, real education and learning are being trivial- ized. We seem to have lost sight of the more significant purpose of teaching and learning: individual growth and develop- ment. We also have surrendered much of the joy of learning, as the incidentals, the tangents, the “bird walks” are cut short or elimi- nated because teachers hear the con- stant ticking clock of the countdown to the state test and feel the pressure of the way-too-many standards that have to be covered in a mere 180 school days. The accountability movement has pushed us away from seeing the whole child: “Students are not machines, as the standards movement suggests; they are volatile, complicated, and paradoxical” (Cookson, 2001, p. 42). How does this impact gifted chil- dren? In many heterogeneous class- rooms, teachers have retreated to traditional subject delineations and traditional instruction in an effort to ensure direct standards-based instruc- tion even though “no solid basis exists in the research literature for the ways we currently develop, place, and align educational standards in school cur- ricula” (Zenger & Zenger, 2002, p. 212). Grade-level standards are often particularly inappropriate for the gifted and talented whose pace of learning, achievement levels, and depth of knowledge are significantly beyond their chronological peers. A broad-based, thematically rich, and challenging curriculum is the heart of education for the gifted. Virgil Ward, one of the earliest voices for a differen- tial education for the gifted, said, “It is insufficient to consider the curriculum for the gifted in terms of traditional subjects and instructional processes” (Ward, 1980, p. 5). VanTassel-Baska Standards-Based v. Standards-Embedded Curriculum gifted child today 45 Standards-Based v. Standards-Embedded Curriculum and Stambaugh (2006) described three dimensions of successful curriculum for gifted students: content mastery, pro- cess and product, and epistemological concept, “understanding and appre- ciating systems of knowledge rather than individual elements of those systems” (p. 9). Overemphasis on testing and grade-level standards limits all three and therefore limits learning for gifted students. Hirsch (2001) concluded that “broad gen- eral knowledge is the best entrée to deep knowledge” (p. 23) and that it is highly correlated with general ability to learn. He continued, “the best way to learn a subject is to learn its gen- eral principles and to study an ample number of diverse examples that illustrate those principles” (Hirsch, 2001, p. 23). Principle-based learn- ing applies to both gifted and general education children. In order to meet the needs of gifted and general education students, cur- riculum should be differentiated in ways that are relevant and engaging. Curriculum content, processes, and products should provide challenge, depth, and complexity, offering multiple opportunities for problem solving, creativity, and exploration. In specific content areas, the cur- riculum should reflect the elegance and sophistication unique to the discipline. Even with this expanded view of curriculum in mind, we still must find ways to address the current reality of state standards and assess- ments. Standards-Embedded Curriculum How can educators address this chal- lenge? As in most things, a change of perspective can be helpful. Standards- based curriculum as described above should be replaced with standards- embedded curriculum. Standards- embedded curriculum begins with broad questions and topics, either discipline specific or interdisciplinary. Once teachers have given thoughtful consideration to relevant, engaging, and important content and the con- nections that support meaning-making (Jensen, 1998), they next select stan- dards that are relevant to this content and to summative assessments. This process is supported by the backward planning advocated in Understanding by Design by Wiggins and McTighe (2005) and its predecessors, as well as current thinkers in other fields, such as Covey (Tomlinson & McTighe, 2006). It is a critical component of differenti- ating instruction for advanced learners (Tomlinson, 2001) and a significant factor in the Core Parallel in the Parallel Curriculum Model (Tomlinson et al., 2002). Teachers choose from standards in multiple disciplines at both above and below grade level depending on the needs of the students and the classroom or program structure. Preassessment data and the results of prior instruc- tion also inform this process of embed- ding appropriate standards. For gifted students, this formative assessment will result in “more advanced curricula available at younger ages, ensuring that all levels of the standards are traversed in the process” (VanTassel-Baska & Little, 2003, p. 3). Once the essential questions, key content, and relevant standards are selected and sequenced, they are embedded into a coherent unit design and instructional decisions (grouping, pacing, instructional methodology) can be made. For gifted students, this includes the identification of appropri- ate resources, often including advanced texts, mentors, and independent research, as appropriate to the child’s developmental level and interest. Applying Standards- Embedded Curriculum What does this look like in practice? In reading the possible class- room applications below, consider these three Ohio Academic Content Standards for third grade: 1. Math: “Read thermometers in both Fahrenheit and Celsius scales” (“Academic Content Standards: K–12 Mathematics,” n.d., p. 71). 2. Social Studies: “Compare some of the cultural practices and products of various groups of people who have lived in the local community including artistic expression, religion, language, and food. Compare the cultural practices and products of the local community with those of other communities in Ohio, the United States, and countries of the world” (Academic Content Standards: K–12 Social Studies, n.d., p. 122). 3. Life Science: “Observe and explore how fossils provide evidence about animals that lived long ago and the nature of the environment at that time” (Academic Content Standards: K–12 Science, n.d., p. 57). When students are fortunate to have a teacher who is dedicated to helping all of them make good use of their time, the gifted may have a preassessment opportunity where they can demonstrate their familiarity with the content and potential mastery of a standard at their grade level. Students who pass may get to read by them- selves for the brief period while the rest of the class works on the single outcome. Sometimes more experienced teachers will create opportunities for gifted and advanced students Standards-Based v. Standards-Embedded Curriculum to work on a standard in the same domain or strand at the next higher grade level (i.e., accelerate through the standards). For example, a stu- dent might be able to work on a Life Science standard for fourth grade that progresses to other communities such as ecosystems. These above-grade-level standards can provide rich material for differentiation, advanced problem solving, and more in-depth curriculum integration. In another classroom scenario, a teacher may focus on the math stan- dard above, identifying the standard number on his lesson plan. He creates or collects paper thermometers, some showing measurement in Celsius and some in Fahrenheit. He also has some real thermometers. He demonstrates thermometer use with boiling water and with freezing water and reads the different temperatures. Students complete a worksheet that has them read thermometers in Celsius and Fahrenheit. The more advanced students may learn how to convert between the two scales. Students then practice with several questions on the topic that are similar in structure and content to those that have been on past proficiency tests. They are coached in how to answer them so that the stan- dard, instruction, formative assess- ment, and summative assessment are all aligned. Then, each student writes a statement that says, “I can read a thermometer using either Celsius or Fahrenheit scales.” Both of these examples describe a standards-based environment, where the starting point is the standard. Direct instruction to that standard is followed by an observable student behavior that demonstrates specific mastery of that single standard. The standard becomes both the start- ing point and the ending point of the curriculum. Education, rather than opening up a student’s mind, becomes a series of closed links in a chain. Whereas the above lessons may be differentiated to some extent, they have no context; they may relate only to the next standard on the list, such as, “Telling time to the nearest minute and finding elapsed time using a cal- endar or a clock.” How would a “standards-embed- ded” model of curriculum design be different? It would begin with the development of an essential ques- tion such as, “Who or what lived here before me? How were they different from me? How were they the same? How do we know?” These questions might be more relevant to our con- temporary highly mobile students. It would involve place and time. Using this intriguing line of inquiry, students might work on the social studies stan- dard as part of the study of their home- town, their school, or even their house or apartment. Because where people live and what they do is influenced by the weather, students could look into weather patterns of their area and learn how to measure temperature using a Fahrenheit scale so they could see if it is similar now to what it was a century ago. Skipping ahead to consideration of the social studies standard, students could then choose another country, preferably one that uses Celsius, and do the same investigation of fossils, communities, and the like. Students could complete a weather comparison, looking at the temperature in Celsius as people in other parts of the world, such as those in Canada, do. Thus, learning is contextualized and connected, dem- onstrating both depth and complexity. This approach takes a lot more work and time. It is a sophisticated integrated view of curriculum devel- opment and involves in-depth knowl- edge of the content areas, as well as an understanding of the scope and sequence of the standards in each dis- cipline. Teachers who develop vital single-discipline units, as well as inter- disciplinary teaching units, begin with a central topic surrounded by subtopics and connections to other areas. Then they connect important terms, facts, or concepts to the subtopics. Next, the skilled teacher/curriculum devel- oper embeds relevant, multileveled standards and objectives appropriate to a given student or group of stu- dents into the unit. Finally, teachers select the instructional strategies and develop student assessments. These assessments include, but are not lim- ited to, the types of questions asked on standardized and state assessments. Comparing Standards- Based and Standards- Embedded Curriculum Design Following is an articulation of the differences between standards-based and standards-embedded curriculum design. (See Figure 1.) 1. The starting point. Standards- based curriculum begins with the grade-level standard and the underlying assumption that every student needs to master that stan- dard at that moment in time. In standards-embedded curriculum, the multifaceted essential ques- tion and students’ needs are the starting points. 2. Preassessment. In standards- based curriculum and teaching, if a preassessment is provided, it cov- ers a single standard or two. In a standards-embedded curriculum, preassessment includes a broader range of grade-level and advanced standards, as well as students’ knowledge of surrounding content such as background experiences with the subject, relevant skills (such as reading and writing), and continued on page ?? even learning style or interests. gifted child today 47 Standards-Based v. Standards-Embedded Curriculum Standards Based Standards Embedded Starting Points The grade-level standard. Whole class’ general skill level Essential questions and content relevant to individual students and groups. Preassessment Targeted to a single grade-level standard. Short-cycle assessments. Background knowledge. Multiple grade-level standards from multiple areas connected by the theme of the unit. Includes annual learning style and interest inventories. Acceleration/ Enrichment To next grade-level standard in the same strand. To above-grade-level standards, as well as into broader thematically connected content. Language Arts Divided into individual skills. Reading and writing skills often separated from real-world relevant contexts. The language arts are embedded in all units and themes and connected to differentiated processes and products across all content areas. Instruction Lesson planning begins with the standard as the objective. Sequential direct instruction progresses through the standards in each content area separately. Strategies are selected to introduce, practice, and demonstrate mastery of all grade-level standards in all content areas in one school year. Lesson planning begins with essential questions, topics, and significant themes. Integrated instruction is designed around connections among content areas and embeds all relevant standards. Assessment Format modeled after the state test. Variety of assessments including questions similar to the state test format. Teacher Role Monitor of standards mastery. Time manager. Facilitator of instructional design and student engagement with learning, as well as assessor of achievement. Student Self- Esteem “I can . . .” statements. Star Charts. Passing “the test.” Completed projects/products. Making personal connections to learning and the theme/topic. Figure 1. Standards based v. standards-embedded instruction and gifted students. and the potential political outcry of “stepping on the toes” of the next grade’s teacher. Few classroom teachers have been provided with the in-depth professional develop- ment and understanding of curric- ulum compacting that would allow them to implement this effectively. In standards-embedded curricu- lum, enrichment and extensions of learning are more possible and more interesting because ideas, top- ics, and questions lend themselves more easily to depth and complex- ity than isolated skills. 4. Language arts. In standards- based classrooms, the language arts have been redivided into sepa- rate skills, with reading separated from writing, and writing sepa- rated from grammar. To many concrete thinkers, whole-language approaches seem antithetical to teaching “to the standards.” In a standards-embedded classroom, integrated language arts skills (reading, writing, listening, speak- ing, presenting, and even pho- nics) are embedded into the study of every unit. Especially for the gifted, the communication and language arts are essential, regard- less of domain-specific talents (Ward, 1980) and should be com- ponents of all curriculum because they are the underpinnings of scholarship in all areas. 5. Instruction. A standards-based classroom lends itself to direct instruction and sequential pro- gression from one standard to the next. A standards-embedded class- room requires a variety of more open-ended instructional strate- gies and materials that extend and diversify learning rather than focus it narrowly. Creativity and differ- entiation in instruction and stu- dent performance are supported more effectively in a standards- embedded approach. 6. Assessment. A standards-based classroom uses targeted assess- ments focused on the structure and content of questions on the externally imposed standardized test (i.e., proficiency tests). A stan- dards-embedded classroom lends itself to greater use of authentic assessment and differentiated 3. Acceleration/Enrichment. In a standards-based curriculum, the narrow definition of the learning outcome (a test item) often makes acceleration or curriculum compact- ing the only path for differentiating instruction for gifted, talented, and/ or advanced learners. This rarely happens, however, because of lack of materials, knowledge, o

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