(a) From Figure 7.25, compute the length of time required for the average grain diameter to increase from 0.03 to 0.3 mm at 600°C for this brass material. (b) Repeat the calculation, this time using 700° C.

(a) From Figure 7.25, compute the length of time required for the average grain diameter to increase from 0.03 to 0.3 mm at 600°C for this brass material. (b) Repeat the calculation, this time using 700° C.

(a) From Figure 7.25, (and realizing that both axes are … Read More...
Silver has two naturally occurring isotopes with the isotopic masses of 106.90509 amu and 108.9047 amu. The average atomic mass of silver is 107.8682 amu. The fractional abundance of the lighter of the two isotopes is __________. A) 0.24221 B) 0.48168 C) 0.51835 D) 0.75783 E) 0.90474

Silver has two naturally occurring isotopes with the isotopic masses of 106.90509 amu and 108.9047 amu. The average atomic mass of silver is 107.8682 amu. The fractional abundance of the lighter of the two isotopes is __________. A) 0.24221 B) 0.48168 C) 0.51835 D) 0.75783 E) 0.90474

C) 0.51835
Assignment 10 Due: 11:59pm on Wednesday, April 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 12.3 Part A The figure shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies to . Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: Ka Kc Correct Conceptual Question 12.6 You have two steel solid spheres. Sphere 2 has twice the radius of sphere 1. Part A By what factor does the moment of inertia of sphere 2 exceed the moment of inertia of sphere 1? ANSWER: I2 I1 Correct Problem 12.2 A high-speed drill reaches 2500 in 0.59 . Part A What is the drill’s angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B Through how many revolutions does it turn during this first 0.59 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct I2/I1 = 32 rpm s  = 440 rad s2 s  = 12 rev Constant Angular Acceleration in the Kitchen Dario, a prep cook at an Italian restaurant, spins a salad spinner and observes that it rotates 20.0 times in 5.00 seconds and then stops spinning it. The salad spinner rotates 6.00 more times before it comes to rest. Assume that the spinner slows down with constant angular acceleration. Part A What is the angular acceleration of the salad spinner as it slows down? Express your answer numerically in degrees per second per second. Hint 1. How to approach the problem Recall from your study of kinematics the three equations of motion derived for systems undergoing constant linear acceleration. You are now studying systems undergoing constant angular acceleration and will need to work with the three analogous equations of motion. Collect your known quantities and then determine which of the angular kinematic equations is appropriate to find the angular acceleration . Hint 2. Find the angular velocity of the salad spinner while Dario is spinning it What is the angular velocity of the salad spinner as Dario is spinning it? Express your answer numerically in degrees per second. Hint 1. Converting rotations to degrees When the salad spinner spins through one revolution, it turns through 360 degrees. ANSWER: Hint 3. Find the angular distance the salad spinner travels as it comes to rest Through how many degrees does the salad spinner rotate as it comes to rest? Express your answer numerically in degrees. Hint 1. Converting rotations to degrees  0 = 1440 degrees/s  =  − 0 One revolution is equivalent to 360 degrees. ANSWER: Hint 4. Determine which equation to use You know the initial and final velocities of the system and the angular distance through which the spinner rotates as it comes to a stop. Which equation should be used to solve for the unknown constant angular acceleration ? ANSWER: ANSWER: Correct Part B How long does it take for the salad spinner to come to rest? Express your answer numerically in seconds.  = 2160 degrees   = 0 + 0t+  1 2 t2  = 0 + t = + 2( − ) 2 20 0  = -480 degrees/s2 Hint 1. How to approach the problem Again, you will need the equations of rotational kinematics that apply to situations of constant angular acceleration. Collect your known quantities and then determine which of the angular kinematic equations is appropriate to find . Hint 2. Determine which equation to use You have the initial and final velocities of the system and the angular acceleration, which you found in the previous part. Which is the best equation to use to solve for the unknown time ? ANSWER: ANSWER: Correct ± A Spinning Electric Fan An electric fan is turned off, and its angular velocity decreases uniformly from 540 to 250 in a time interval of length 4.40 . Part A Find the angular acceleration in revolutions per second per second. Hint 1. Average acceleration Recall that if the angular velocity decreases uniformly, the angular acceleration will remain constant. Therefore, the angular acceleration is just the total change in angular velocity divided by t t  = 0 + 0t+  1 2 t2  = 0 + t = + 2( − ) 2 20 0 t = 3.00 s rev/min rev/min s  the total change in time. Be careful of the sign of the angular acceleration. ANSWER: Correct Part B Find the number of revolutions made by the fan blades during the time that they are slowing down in Part A. Hint 1. Determine the correct kinematic equation Which of the following kinematic equations is best suited to this problem? Here and are the initial and final angular velocities, is the elapsed time, is the constant angular acceleration, and and are the initial and final angular displacements. Hint 1. How to chose the right equation Notice that you were given in the problem introduction the initial and final speeds, as well as the length of time between them. In this problem, you are asked to find the number of revolutions (which here is the change in angular displacement, ). If you already found the angular acceleration in Part A, you could use that as well, but you would end up using a more complex equation. Also, in general, it is somewhat favorable to use given quantities instead of quantities that you have calculated. ANSWER:  = -1.10 rev/s2 0  t  0   − 0  = 0 + t  = 0 + t+  1 2 t2 = + 2( − ) 2 20 0 − 0 = (+ )t 1 2 0 ANSWER: Correct Part C How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in Part A? Hint 1. Finding the total time for spin down To find the total time for spin down, just calculate when the velocity will equal zero. This is accomplished by setting the initial velocity plus the acceleration multipled by the time equal to zero and then solving for the time. One can then just subtract the time it took to reach 250 from the total time. Be careful of your signs when you set up the equation. ANSWER: Correct Problem 12.8 A 100 ball and a 230 ball are connected by a 34- -long, massless, rigid rod. The balls rotate about their center of mass at 130 . Part A What is the speed of the 100 ball? Express your answer to two significant figures and include the appropriate units. ANSWER: 29.0 rev rev/min 3.79 s g g cm rpm g Correct Problem 12.10 A thin, 60.0 disk with a diameter of 9.00 rotates about an axis through its center with 0.200 of kinetic energy. Part A What is the speed of a point on the rim? Express your answer with the appropriate units. ANSWER: Correct Problem 12.12 A drum major twirls a 95- -long, 470 baton about its center of mass at 150 . Part A What is the baton’s rotational kinetic energy? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 3.2 ms g cm J 3.65 ms cm g rpm K = 4.4 J Correct Net Torque on a Pulley The figure below shows two blocks suspended by a cord over a pulley. The mass of block B is twice the mass of block A, while the mass of the pulley is equal to the mass of block A. The blocks are let free to move and the cord moves on the pulley without slipping or stretching. There is no friction in the pulley axle, and the cord’s weight can be ignored. Part A Which of the following statements correctly describes the system shown in the figure? Check all that apply. Hint 1. Conditions for equilibrium If the blocks had the same mass, the system would be in equilibrium. The blocks would have zero acceleration and the tension in each part of the cord would equal the weight of each block. Both parts of the cord would then pull with equal force on the pulley, resulting in a zero net torque and no rotation of the pulley. Is this still the case in the current situation where block B has twice the mass of block A? Hint 2. Rotational analogue of Newton’s second law The net torque of all the forces acting on a rigid body is proportional to the angular acceleration of the body net  and is given by , where is the moment of inertia of the body. Hint 3. Relation between linear and angular acceleration A particle that rotates with angular acceleration has linear acceleration equal to , where is the distance of the particle from the axis of rotation. In the present case, where there is no slipping or stretching of the cord, the cord and the pulley must move together at the same speed. Therefore, if the cord moves with linear acceleration , the pulley must rotate with angular acceleration , where is the radius of the pulley. ANSWER: Correct Part B What happens when block B moves downward? Hint 1. How to approach the problem To determine whether the tensions in both parts of the cord are equal, it is convenient to write a mathematical expression for the net torque on the pulley. This will allow you to relate the tensions in the cord to the pulley’s angular acceleration. Hint 2. Find the net torque on the pulley Let’s assume that the tensions in both parts of the cord are different. Let be the tension in the right cord and the tension in the left cord. If is the radius of the pulley, what is the net torque acting on the pulley? Take the positive sense of rotation to be counterclockwise. Express your answer in terms of , , and . net = I I  a a = R R a  = a R R The acceleration of the blocks is zero. The net torque on the pulley is zero. The angular acceleration of the pulley is nonzero. T1 T2 R net T1 T2 R Hint 1. Torque The torque of a force with respect to a point is defined as the product of the magnitude times the perpendicular distance between the line of action of and the point . In other words, . ANSWER: ANSWER: Correct Note that if the pulley were stationary (as in many systems where only linear motion is studied), then the tensions in both parts of the cord would be equal. However, if the pulley rotates with a certain angular acceleration, as in the present situation, the tensions must be different. If they were equal, the pulley could not have an angular acceleration. Problem 12.18 Part A In the figure , what is the magnitude of net torque about the axle? Express your answer to two significant figures and include the appropriate units.  F  O F l F  O  = Fl net = R(T2 − T1 ) The left cord pulls on the pulley with greater force than the right cord. The left and right cord pull with equal force on the pulley. The right cord pulls on the pulley with greater force than the left cord. ANSWER: Correct Part B What is the direction of net torque about the axle? ANSWER: Correct Problem 12.22 An athlete at the gym holds a 3.5 steel ball in his hand. His arm is 78 long and has a mass of 3.6 . Assume the center of mass of the arm is at the geometrical center of the arm. Part A What is the magnitude of the torque about his shoulder if he holds his arm straight out to his side, parallel to the floor? Express your answer to two significant figures and include the appropriate units.  = 0.20 Nm Clockwise Counterclockwise kg cm kg ANSWER: Correct Part B What is the magnitude of the torque about his shoulder if he holds his arm straight, but below horizontal? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Parallel Axis Theorem The parallel axis theorem relates , the moment of inertia of an object about an axis passing through its center of mass, to , the moment of inertia of the same object about a parallel axis passing through point p. The mathematical statement of the theorem is , where is the perpendicular distance from the center of mass to the axis that passes through point p, and is the mass of the object. Part A Suppose a uniform slender rod has length and mass . The moment of inertia of the rod about about an axis that is perpendicular to the rod and that passes through its center of mass is given by . Find , the moment of inertia of the rod with respect to a parallel axis through one end of the rod. Express in terms of and . Use fractions rather than decimal numbers in your answer. Hint 1. Find the distance from the axis to the center of mass Find the distance appropriate to this problem. That is, find the perpendicular distance from the center of mass of the rod to the axis passing through one end of the rod.  = 41 Nm 45  = 29 Nm Icm Ip Ip = Icm + Md2 d M L m Icm = m 1 12 L2 Iend Iend m L d ANSWER: ANSWER: Correct Part B Now consider a cube of mass with edges of length . The moment of inertia of the cube about an axis through its center of mass and perpendicular to one of its faces is given by . Find , the moment of inertia about an axis p through one of the edges of the cube Express in terms of and . Use fractions rather than decimal numbers in your answer. Hint 1. Find the distance from the axis to the axis Find the perpendicular distance from the center of mass axis to the new edge axis (axis labeled p in the figure). ANSWER: d = L 2 Iend = mL2 3 m a Icm Icm = m 1 6 a2 Iedge Iedge m a o p d ANSWER: Correct Problem 12.26 Starting from rest, a 12- -diameter compact disk takes 2.9 to reach its operating angular velocity of 2000 . Assume that the angular acceleration is constant. The disk’s moment of inertia is . Part A How much torque is applied to the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How many revolutions does it make before reaching full speed? Express your answer using two significant figures. ANSWER: d = a 2 Iedge = 2ma2 3 cm s rpm 2.5 × 10−5 kg m2 = 1.8×10−3  Nm Correct Problem 12.23 An object’s moment of inertia is 2.20 . Its angular velocity is increasing at the rate of 3.70 . Part A What is the total torque on the object? ANSWER: Correct Problem 12.31 A 5.1 cat and a 2.5 bowl of tuna fish are at opposite ends of the 4.0- -long seesaw. N = 48 rev kgm2 rad/s2 8.14 N  m kg kg m Part A How far to the left of the pivot must a 3.8 cat stand to keep the seesaw balanced? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Static Equilibrium of the Arm You are able to hold out your arm in an outstretched horizontal position because of the action of the deltoid muscle. Assume the humerus bone has a mass , length and its center of mass is a distance from the scapula. (For this problem ignore the rest of the arm.) The deltoid muscle attaches to the humerus a distance from the scapula. The deltoid muscle makes an angle of with the horizontal, as shown. Use throughout the problem. Part A kg d = 1.4 m M1 = 3.6 kg L = 0.66 m L1 = 0.33 m L2 = 0.15 m  = 17 g = 9.8 m/s2 Find the tension in the deltoid muscle. Express the tension in newtons, to the nearest integer. Hint 1. Nature of the problem Remember that this is a statics problem, so all forces and torques are balanced (their sums equal zero). Hint 2. Origin of torque Calculate the torque about the point at which the arm attaches to the rest of the body. This allows one to balance the torques without having to worry about the undefined forces at this point. Hint 3. Adding up the torques Add up the torques about the point in which the humerus attaches to the body. Answer in terms of , , , , , and . Remember that counterclockwise torque is positive. ANSWER: ANSWER: Correct Part B Using the conditions for static equilibrium, find the magnitude of the vertical component of the force exerted by the scapula on the humerus (where the humerus attaches to the rest of the body). Express your answer in newtons, to the nearest integer. T L1 L2 M1 g T  total = 0 = L1M1g − Tsin()L2 T = 265 N Fy Hint 1. Total forces involved Recall that there are three vertical forces in this problem: the force of gravity acting on the bone, the force from the vertical component of the muscle tension, and the force exerted by the scapula on the humerus (where it attaches to the rest of the body). ANSWER: Correct Part C Now find the magnitude of the horizontal component of the force exerted by the scapula on the humerus. Express your answer in newtons, to the nearest integer. ANSWER: Correct ± Moments around a Rod A rod is bent into an L shape and attached at one point to a pivot. The rod sits on a frictionless table and the diagram is a view from above. This means that gravity can be ignored for this problem. There are three forces that are applied to the rod at different points and angles: , , and . Note that the dimensions of the bent rod are in centimeters in the figure, although the answers are requested in SI units (kilograms, meters, seconds). |Fy| = 42 N Fx |Fx| = 254 N F 1 F  2 F  3 Part A If and , what does the magnitude of have to be for there to be rotational equilibrium? Answer numerically in newtons to two significant figures. Hint 1. Finding torque about pivot from What is the magnitude of the torque | | provided by around the pivot point? Give your answer numerically in newton-meters to two significant figures. ANSWER: ANSWER: Correct Part B If the L-shaped rod has a moment of inertia , , , and again , how long a time would it take for the object to move through ( /4 radians)? Assume that as the object starts to move, each force moves with the object so as to retain its initial angle relative to the object. Express the time in seconds to two significant figures. F3 = 0 F1 = 12 N F 2 F 1   1 F  1 |  1 | = 0.36 N  m F2 = 4.5 N I = 9 kg m2 F1 = 12 N F2 = 27 N F3 = 0 t 45  Hint 1. Find the net torque about the pivot What is the magnitude of the total torque around the pivot point? Answer numerically in newton-meters to two significant figures. ANSWER: Hint 2. Calculate Given the total torque around the pivot point, what is , the magnitude of the angular acceleration? Express your answer numerically in radians per second squared to two significant figures. Hint 1. Equation for If you know the magnitude of the total torque ( ) and the rotational inertia ( ), you can then find the rotational acceleration ( ) from ANSWER: Hint 3. Description of angular kinematics Now that you know the angular acceleration, this is a problem in rotational kinematics; find the time needed to go through a given angle . For constant acceleration ( ) and starting with (where is angular speed) the relation is given by which is analogous to the expression for linear displacement ( ) with constant acceleration ( ) starting from rest, | p ivot| | p ivot| = 1.8 N  m    vot Ivot  pivot = Ipivot.  = 0.20 radians/s2    = 0   = 1  , 2 t2 x a . ANSWER: Correct Part C Now consider the situation in which and , but now a force with nonzero magnitude is acting on the rod. What does have to be to obtain equilibrium? Give a numerical answer, without trigonometric functions, in newtons, to two significant figures. Hint 1. Find the required component of Only the tangential (perpendicular) component of (call it ) provides a torque. What is ? Answer in terms of . You will need to evaluate any trigonometric functions. ANSWER: ANSWER: Correct x = 1 a 2 t2 t = 2.8 s F1 = 12 N F2 = 0 F3 F3 F 3 F  3 F3t F3t F3 F3t = 1 2 F3 F3 = 9.0 N Problem 12.32 A car tire is 55.0 in diameter. The car is traveling at a speed of 24.0 . Part A What is the tire’s rotation frequency, in rpm? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B What is the speed of a point at the top edge of the tire? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C What is the speed of a point at the bottom edge of the tire? Express your answer as an integer and include the appropriate units. ANSWER: cm m/s 833 rpm 48.0 ms 0 ms Correct Problem 12.33 A 460 , 8.00-cm-diameter solid cylinder rolls across the floor at 1.30 . Part A What is the can’s kinetic energy? Express your answer with the appropriate units. ANSWER: Correct Problem 12.45 Part A What is the magnitude of the angular momentum of the 780 rotating bar in the figure ? g m/s 0.583 J g ANSWER: Correct Part B What is the direction of the angular momentum of the bar ? ANSWER: Correct Problem 12.46 Part A What is the magnitude of the angular momentum of the 2.20 , 4.60-cm-diameter rotating disk in the figure ? 3.27 kgm2/s into the page out of the page kg ANSWER: Correct Part B What is its direction? ANSWER: Correct Problem 12.60 A 3.0- -long ladder, as shown in the following figure, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.46. 3.66×10−2 kgm /s 2 x direction -x direction y direction -y direction z direction -z direction m Part A What is the minimum angle the ladder can make with the floor without slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.61 The 3.0- -long, 90 rigid beam in the following figure is supported at each end. An 70 student stands 2.0 from support 1.  = 47 m kg kg m Part A How much upward force does the support 1 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How much upward force does the support 2 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 12.63 A 44 , 5.5- -long beam is supported, but not attached to, the two posts in the figure . A 22 boy starts walking along the beam. You may want to review ( pages 330 – 334) . For help with math skills, you may want to review: F1 = 670 N F2 = 900 N kg m kg The Vector Cross Product Part A How close can he get to the right end of the beam without it falling over? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture of the four forces acting on the beam, indicating both their direction and the place on the beam that the forces are acting. Choose a coordinate system with a direction for the axis along the beam, and indicate the position of the boy. What is the net force on the beam if it is stationary? Just before the beam tips, the force of the left support on the beam is zero. Using the zero net force condition, what is the force due to the right support just before the beam tips? For the beam to remain stationary, what must be zero besides the net force on the beam? Choose a point on the beam, and compute the net torque on the beam about that point. Be sure to choose a positive direction for the rotation axis and therefore the torques. Using the zero torque condition, what is the position of the boy on the beam just prior to tipping? How far is this position from the right edge of the beam? ANSWER: Correct d = 2.0 m Problem 12.68 Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.6 diameter and a mass of 270 . Its maximum angular velocity is 1500 . Part A A motor spins up the flywheel with a constant torque of 54 . How long does it take the flywheel to reach top speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How much energy is stored in the flywheel? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.2 . What is the average power delivered to the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: m kg rpm N  m t = 250 s = 1.1×106 E J s Correct Part D How much torque does the flywheel exert on the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.71 The 3.30 , 40.0-cm-diameter disk in the figure is spinning at 350 . Part A How much friction force must the brake apply to the rim to bring the disk to a halt in 2.10 ? P = 2.4×105 W  = 1800 Nm kg rpm s Express your answer with the appropriate units. ANSWER: Correct Problem 12.74 A 5.0 , 60- -diameter cylinder rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. Part A What is the magnitude of the cylinder’s initial angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: 5.76 N kg cm  = 22 rad s2 Correct Part B What is the magnitude of the cylinder’s angular velocity when it is directly below the axle? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.82 A 45 figure skater is spinning on the toes of her skates at 0.90 . Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 , 20 average diameter, 160 tall) plus two rod-like arms (2.5 each, 67 long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 , 20- -diameter, 200- -tall cylinder. Part A What is her new rotation frequency, in revolutions per second? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Score Summary:  = 6.6 rad s kg rev/s kg cm cm kg cm kg cm cm 2 = Your score on this assignment is 95.7%. You received 189.42 out of a possible total of 198 points.

Assignment 10 Due: 11:59pm on Wednesday, April 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 12.3 Part A The figure shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies to . Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: Ka Kc Correct Conceptual Question 12.6 You have two steel solid spheres. Sphere 2 has twice the radius of sphere 1. Part A By what factor does the moment of inertia of sphere 2 exceed the moment of inertia of sphere 1? ANSWER: I2 I1 Correct Problem 12.2 A high-speed drill reaches 2500 in 0.59 . Part A What is the drill’s angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B Through how many revolutions does it turn during this first 0.59 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct I2/I1 = 32 rpm s  = 440 rad s2 s  = 12 rev Constant Angular Acceleration in the Kitchen Dario, a prep cook at an Italian restaurant, spins a salad spinner and observes that it rotates 20.0 times in 5.00 seconds and then stops spinning it. The salad spinner rotates 6.00 more times before it comes to rest. Assume that the spinner slows down with constant angular acceleration. Part A What is the angular acceleration of the salad spinner as it slows down? Express your answer numerically in degrees per second per second. Hint 1. How to approach the problem Recall from your study of kinematics the three equations of motion derived for systems undergoing constant linear acceleration. You are now studying systems undergoing constant angular acceleration and will need to work with the three analogous equations of motion. Collect your known quantities and then determine which of the angular kinematic equations is appropriate to find the angular acceleration . Hint 2. Find the angular velocity of the salad spinner while Dario is spinning it What is the angular velocity of the salad spinner as Dario is spinning it? Express your answer numerically in degrees per second. Hint 1. Converting rotations to degrees When the salad spinner spins through one revolution, it turns through 360 degrees. ANSWER: Hint 3. Find the angular distance the salad spinner travels as it comes to rest Through how many degrees does the salad spinner rotate as it comes to rest? Express your answer numerically in degrees. Hint 1. Converting rotations to degrees  0 = 1440 degrees/s  =  − 0 One revolution is equivalent to 360 degrees. ANSWER: Hint 4. Determine which equation to use You know the initial and final velocities of the system and the angular distance through which the spinner rotates as it comes to a stop. Which equation should be used to solve for the unknown constant angular acceleration ? ANSWER: ANSWER: Correct Part B How long does it take for the salad spinner to come to rest? Express your answer numerically in seconds.  = 2160 degrees   = 0 + 0t+  1 2 t2  = 0 + t = + 2( − ) 2 20 0  = -480 degrees/s2 Hint 1. How to approach the problem Again, you will need the equations of rotational kinematics that apply to situations of constant angular acceleration. Collect your known quantities and then determine which of the angular kinematic equations is appropriate to find . Hint 2. Determine which equation to use You have the initial and final velocities of the system and the angular acceleration, which you found in the previous part. Which is the best equation to use to solve for the unknown time ? ANSWER: ANSWER: Correct ± A Spinning Electric Fan An electric fan is turned off, and its angular velocity decreases uniformly from 540 to 250 in a time interval of length 4.40 . Part A Find the angular acceleration in revolutions per second per second. Hint 1. Average acceleration Recall that if the angular velocity decreases uniformly, the angular acceleration will remain constant. Therefore, the angular acceleration is just the total change in angular velocity divided by t t  = 0 + 0t+  1 2 t2  = 0 + t = + 2( − ) 2 20 0 t = 3.00 s rev/min rev/min s  the total change in time. Be careful of the sign of the angular acceleration. ANSWER: Correct Part B Find the number of revolutions made by the fan blades during the time that they are slowing down in Part A. Hint 1. Determine the correct kinematic equation Which of the following kinematic equations is best suited to this problem? Here and are the initial and final angular velocities, is the elapsed time, is the constant angular acceleration, and and are the initial and final angular displacements. Hint 1. How to chose the right equation Notice that you were given in the problem introduction the initial and final speeds, as well as the length of time between them. In this problem, you are asked to find the number of revolutions (which here is the change in angular displacement, ). If you already found the angular acceleration in Part A, you could use that as well, but you would end up using a more complex equation. Also, in general, it is somewhat favorable to use given quantities instead of quantities that you have calculated. ANSWER:  = -1.10 rev/s2 0  t  0   − 0  = 0 + t  = 0 + t+  1 2 t2 = + 2( − ) 2 20 0 − 0 = (+ )t 1 2 0 ANSWER: Correct Part C How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in Part A? Hint 1. Finding the total time for spin down To find the total time for spin down, just calculate when the velocity will equal zero. This is accomplished by setting the initial velocity plus the acceleration multipled by the time equal to zero and then solving for the time. One can then just subtract the time it took to reach 250 from the total time. Be careful of your signs when you set up the equation. ANSWER: Correct Problem 12.8 A 100 ball and a 230 ball are connected by a 34- -long, massless, rigid rod. The balls rotate about their center of mass at 130 . Part A What is the speed of the 100 ball? Express your answer to two significant figures and include the appropriate units. ANSWER: 29.0 rev rev/min 3.79 s g g cm rpm g Correct Problem 12.10 A thin, 60.0 disk with a diameter of 9.00 rotates about an axis through its center with 0.200 of kinetic energy. Part A What is the speed of a point on the rim? Express your answer with the appropriate units. ANSWER: Correct Problem 12.12 A drum major twirls a 95- -long, 470 baton about its center of mass at 150 . Part A What is the baton’s rotational kinetic energy? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 3.2 ms g cm J 3.65 ms cm g rpm K = 4.4 J Correct Net Torque on a Pulley The figure below shows two blocks suspended by a cord over a pulley. The mass of block B is twice the mass of block A, while the mass of the pulley is equal to the mass of block A. The blocks are let free to move and the cord moves on the pulley without slipping or stretching. There is no friction in the pulley axle, and the cord’s weight can be ignored. Part A Which of the following statements correctly describes the system shown in the figure? Check all that apply. Hint 1. Conditions for equilibrium If the blocks had the same mass, the system would be in equilibrium. The blocks would have zero acceleration and the tension in each part of the cord would equal the weight of each block. Both parts of the cord would then pull with equal force on the pulley, resulting in a zero net torque and no rotation of the pulley. Is this still the case in the current situation where block B has twice the mass of block A? Hint 2. Rotational analogue of Newton’s second law The net torque of all the forces acting on a rigid body is proportional to the angular acceleration of the body net  and is given by , where is the moment of inertia of the body. Hint 3. Relation between linear and angular acceleration A particle that rotates with angular acceleration has linear acceleration equal to , where is the distance of the particle from the axis of rotation. In the present case, where there is no slipping or stretching of the cord, the cord and the pulley must move together at the same speed. Therefore, if the cord moves with linear acceleration , the pulley must rotate with angular acceleration , where is the radius of the pulley. ANSWER: Correct Part B What happens when block B moves downward? Hint 1. How to approach the problem To determine whether the tensions in both parts of the cord are equal, it is convenient to write a mathematical expression for the net torque on the pulley. This will allow you to relate the tensions in the cord to the pulley’s angular acceleration. Hint 2. Find the net torque on the pulley Let’s assume that the tensions in both parts of the cord are different. Let be the tension in the right cord and the tension in the left cord. If is the radius of the pulley, what is the net torque acting on the pulley? Take the positive sense of rotation to be counterclockwise. Express your answer in terms of , , and . net = I I  a a = R R a  = a R R The acceleration of the blocks is zero. The net torque on the pulley is zero. The angular acceleration of the pulley is nonzero. T1 T2 R net T1 T2 R Hint 1. Torque The torque of a force with respect to a point is defined as the product of the magnitude times the perpendicular distance between the line of action of and the point . In other words, . ANSWER: ANSWER: Correct Note that if the pulley were stationary (as in many systems where only linear motion is studied), then the tensions in both parts of the cord would be equal. However, if the pulley rotates with a certain angular acceleration, as in the present situation, the tensions must be different. If they were equal, the pulley could not have an angular acceleration. Problem 12.18 Part A In the figure , what is the magnitude of net torque about the axle? Express your answer to two significant figures and include the appropriate units.  F  O F l F  O  = Fl net = R(T2 − T1 ) The left cord pulls on the pulley with greater force than the right cord. The left and right cord pull with equal force on the pulley. The right cord pulls on the pulley with greater force than the left cord. ANSWER: Correct Part B What is the direction of net torque about the axle? ANSWER: Correct Problem 12.22 An athlete at the gym holds a 3.5 steel ball in his hand. His arm is 78 long and has a mass of 3.6 . Assume the center of mass of the arm is at the geometrical center of the arm. Part A What is the magnitude of the torque about his shoulder if he holds his arm straight out to his side, parallel to the floor? Express your answer to two significant figures and include the appropriate units.  = 0.20 Nm Clockwise Counterclockwise kg cm kg ANSWER: Correct Part B What is the magnitude of the torque about his shoulder if he holds his arm straight, but below horizontal? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Parallel Axis Theorem The parallel axis theorem relates , the moment of inertia of an object about an axis passing through its center of mass, to , the moment of inertia of the same object about a parallel axis passing through point p. The mathematical statement of the theorem is , where is the perpendicular distance from the center of mass to the axis that passes through point p, and is the mass of the object. Part A Suppose a uniform slender rod has length and mass . The moment of inertia of the rod about about an axis that is perpendicular to the rod and that passes through its center of mass is given by . Find , the moment of inertia of the rod with respect to a parallel axis through one end of the rod. Express in terms of and . Use fractions rather than decimal numbers in your answer. Hint 1. Find the distance from the axis to the center of mass Find the distance appropriate to this problem. That is, find the perpendicular distance from the center of mass of the rod to the axis passing through one end of the rod.  = 41 Nm 45  = 29 Nm Icm Ip Ip = Icm + Md2 d M L m Icm = m 1 12 L2 Iend Iend m L d ANSWER: ANSWER: Correct Part B Now consider a cube of mass with edges of length . The moment of inertia of the cube about an axis through its center of mass and perpendicular to one of its faces is given by . Find , the moment of inertia about an axis p through one of the edges of the cube Express in terms of and . Use fractions rather than decimal numbers in your answer. Hint 1. Find the distance from the axis to the axis Find the perpendicular distance from the center of mass axis to the new edge axis (axis labeled p in the figure). ANSWER: d = L 2 Iend = mL2 3 m a Icm Icm = m 1 6 a2 Iedge Iedge m a o p d ANSWER: Correct Problem 12.26 Starting from rest, a 12- -diameter compact disk takes 2.9 to reach its operating angular velocity of 2000 . Assume that the angular acceleration is constant. The disk’s moment of inertia is . Part A How much torque is applied to the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How many revolutions does it make before reaching full speed? Express your answer using two significant figures. ANSWER: d = a 2 Iedge = 2ma2 3 cm s rpm 2.5 × 10−5 kg m2 = 1.8×10−3  Nm Correct Problem 12.23 An object’s moment of inertia is 2.20 . Its angular velocity is increasing at the rate of 3.70 . Part A What is the total torque on the object? ANSWER: Correct Problem 12.31 A 5.1 cat and a 2.5 bowl of tuna fish are at opposite ends of the 4.0- -long seesaw. N = 48 rev kgm2 rad/s2 8.14 N  m kg kg m Part A How far to the left of the pivot must a 3.8 cat stand to keep the seesaw balanced? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Static Equilibrium of the Arm You are able to hold out your arm in an outstretched horizontal position because of the action of the deltoid muscle. Assume the humerus bone has a mass , length and its center of mass is a distance from the scapula. (For this problem ignore the rest of the arm.) The deltoid muscle attaches to the humerus a distance from the scapula. The deltoid muscle makes an angle of with the horizontal, as shown. Use throughout the problem. Part A kg d = 1.4 m M1 = 3.6 kg L = 0.66 m L1 = 0.33 m L2 = 0.15 m  = 17 g = 9.8 m/s2 Find the tension in the deltoid muscle. Express the tension in newtons, to the nearest integer. Hint 1. Nature of the problem Remember that this is a statics problem, so all forces and torques are balanced (their sums equal zero). Hint 2. Origin of torque Calculate the torque about the point at which the arm attaches to the rest of the body. This allows one to balance the torques without having to worry about the undefined forces at this point. Hint 3. Adding up the torques Add up the torques about the point in which the humerus attaches to the body. Answer in terms of , , , , , and . Remember that counterclockwise torque is positive. ANSWER: ANSWER: Correct Part B Using the conditions for static equilibrium, find the magnitude of the vertical component of the force exerted by the scapula on the humerus (where the humerus attaches to the rest of the body). Express your answer in newtons, to the nearest integer. T L1 L2 M1 g T  total = 0 = L1M1g − Tsin()L2 T = 265 N Fy Hint 1. Total forces involved Recall that there are three vertical forces in this problem: the force of gravity acting on the bone, the force from the vertical component of the muscle tension, and the force exerted by the scapula on the humerus (where it attaches to the rest of the body). ANSWER: Correct Part C Now find the magnitude of the horizontal component of the force exerted by the scapula on the humerus. Express your answer in newtons, to the nearest integer. ANSWER: Correct ± Moments around a Rod A rod is bent into an L shape and attached at one point to a pivot. The rod sits on a frictionless table and the diagram is a view from above. This means that gravity can be ignored for this problem. There are three forces that are applied to the rod at different points and angles: , , and . Note that the dimensions of the bent rod are in centimeters in the figure, although the answers are requested in SI units (kilograms, meters, seconds). |Fy| = 42 N Fx |Fx| = 254 N F 1 F  2 F  3 Part A If and , what does the magnitude of have to be for there to be rotational equilibrium? Answer numerically in newtons to two significant figures. Hint 1. Finding torque about pivot from What is the magnitude of the torque | | provided by around the pivot point? Give your answer numerically in newton-meters to two significant figures. ANSWER: ANSWER: Correct Part B If the L-shaped rod has a moment of inertia , , , and again , how long a time would it take for the object to move through ( /4 radians)? Assume that as the object starts to move, each force moves with the object so as to retain its initial angle relative to the object. Express the time in seconds to two significant figures. F3 = 0 F1 = 12 N F 2 F 1   1 F  1 |  1 | = 0.36 N  m F2 = 4.5 N I = 9 kg m2 F1 = 12 N F2 = 27 N F3 = 0 t 45  Hint 1. Find the net torque about the pivot What is the magnitude of the total torque around the pivot point? Answer numerically in newton-meters to two significant figures. ANSWER: Hint 2. Calculate Given the total torque around the pivot point, what is , the magnitude of the angular acceleration? Express your answer numerically in radians per second squared to two significant figures. Hint 1. Equation for If you know the magnitude of the total torque ( ) and the rotational inertia ( ), you can then find the rotational acceleration ( ) from ANSWER: Hint 3. Description of angular kinematics Now that you know the angular acceleration, this is a problem in rotational kinematics; find the time needed to go through a given angle . For constant acceleration ( ) and starting with (where is angular speed) the relation is given by which is analogous to the expression for linear displacement ( ) with constant acceleration ( ) starting from rest, | p ivot| | p ivot| = 1.8 N  m    vot Ivot  pivot = Ipivot.  = 0.20 radians/s2    = 0   = 1  , 2 t2 x a . ANSWER: Correct Part C Now consider the situation in which and , but now a force with nonzero magnitude is acting on the rod. What does have to be to obtain equilibrium? Give a numerical answer, without trigonometric functions, in newtons, to two significant figures. Hint 1. Find the required component of Only the tangential (perpendicular) component of (call it ) provides a torque. What is ? Answer in terms of . You will need to evaluate any trigonometric functions. ANSWER: ANSWER: Correct x = 1 a 2 t2 t = 2.8 s F1 = 12 N F2 = 0 F3 F3 F 3 F  3 F3t F3t F3 F3t = 1 2 F3 F3 = 9.0 N Problem 12.32 A car tire is 55.0 in diameter. The car is traveling at a speed of 24.0 . Part A What is the tire’s rotation frequency, in rpm? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part B What is the speed of a point at the top edge of the tire? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C What is the speed of a point at the bottom edge of the tire? Express your answer as an integer and include the appropriate units. ANSWER: cm m/s 833 rpm 48.0 ms 0 ms Correct Problem 12.33 A 460 , 8.00-cm-diameter solid cylinder rolls across the floor at 1.30 . Part A What is the can’s kinetic energy? Express your answer with the appropriate units. ANSWER: Correct Problem 12.45 Part A What is the magnitude of the angular momentum of the 780 rotating bar in the figure ? g m/s 0.583 J g ANSWER: Correct Part B What is the direction of the angular momentum of the bar ? ANSWER: Correct Problem 12.46 Part A What is the magnitude of the angular momentum of the 2.20 , 4.60-cm-diameter rotating disk in the figure ? 3.27 kgm2/s into the page out of the page kg ANSWER: Correct Part B What is its direction? ANSWER: Correct Problem 12.60 A 3.0- -long ladder, as shown in the following figure, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.46. 3.66×10−2 kgm /s 2 x direction -x direction y direction -y direction z direction -z direction m Part A What is the minimum angle the ladder can make with the floor without slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.61 The 3.0- -long, 90 rigid beam in the following figure is supported at each end. An 70 student stands 2.0 from support 1.  = 47 m kg kg m Part A How much upward force does the support 1 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How much upward force does the support 2 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 12.63 A 44 , 5.5- -long beam is supported, but not attached to, the two posts in the figure . A 22 boy starts walking along the beam. You may want to review ( pages 330 – 334) . For help with math skills, you may want to review: F1 = 670 N F2 = 900 N kg m kg The Vector Cross Product Part A How close can he get to the right end of the beam without it falling over? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture of the four forces acting on the beam, indicating both their direction and the place on the beam that the forces are acting. Choose a coordinate system with a direction for the axis along the beam, and indicate the position of the boy. What is the net force on the beam if it is stationary? Just before the beam tips, the force of the left support on the beam is zero. Using the zero net force condition, what is the force due to the right support just before the beam tips? For the beam to remain stationary, what must be zero besides the net force on the beam? Choose a point on the beam, and compute the net torque on the beam about that point. Be sure to choose a positive direction for the rotation axis and therefore the torques. Using the zero torque condition, what is the position of the boy on the beam just prior to tipping? How far is this position from the right edge of the beam? ANSWER: Correct d = 2.0 m Problem 12.68 Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.6 diameter and a mass of 270 . Its maximum angular velocity is 1500 . Part A A motor spins up the flywheel with a constant torque of 54 . How long does it take the flywheel to reach top speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How much energy is stored in the flywheel? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.2 . What is the average power delivered to the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: m kg rpm N  m t = 250 s = 1.1×106 E J s Correct Part D How much torque does the flywheel exert on the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.71 The 3.30 , 40.0-cm-diameter disk in the figure is spinning at 350 . Part A How much friction force must the brake apply to the rim to bring the disk to a halt in 2.10 ? P = 2.4×105 W  = 1800 Nm kg rpm s Express your answer with the appropriate units. ANSWER: Correct Problem 12.74 A 5.0 , 60- -diameter cylinder rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. Part A What is the magnitude of the cylinder’s initial angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: 5.76 N kg cm  = 22 rad s2 Correct Part B What is the magnitude of the cylinder’s angular velocity when it is directly below the axle? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 12.82 A 45 figure skater is spinning on the toes of her skates at 0.90 . Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 , 20 average diameter, 160 tall) plus two rod-like arms (2.5 each, 67 long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 , 20- -diameter, 200- -tall cylinder. Part A What is her new rotation frequency, in revolutions per second? Express your answer to two significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Score Summary:  = 6.6 rad s kg rev/s kg cm cm kg cm kg cm cm 2 = Your score on this assignment is 95.7%. You received 189.42 out of a possible total of 198 points.

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EGR 140 Scientific Programming Assignment # 7 Spring 2017 Use MATLAB to solve each problem by writing script files; copy and paste the script file AND the results in the Command Window and/or plot in the Figure Window to a WORD document that has your name and section in the headers of each page and the page number in each footer. Edit the output to remove extra lines and empty spaces. The script files SHOULD have comments for easy readability; take a print out of the Word file and staple before submission. Due by 3 PM on April 11, 2017. Write a used-defined function that calculates the average and the standard deviation of a list of numbers. Use the function to calculate the average and the standard deviation of the following list of grades: 80 75 91 60 79 89 65 80 95 50 81 Note: The average x_ave (or mean) of a given set of n number x_1,x_2,…..,x_n is given by: x_ave=(x_1+x_2+x_3+⋯+x_n)/n The standard deviation is given by: σ=√((∑_(i=1)^(i=n)▒(x_i-x_ave )^2 )/(n-1)) DO not use built-in functions to calculate the mean and the standard deviation. Write a user-defined function that arranges the digits of a given (positive) number in a row vector in the same order as they appear in the number; the function should also arrange the digits in the decimal part in a different vector. For example, if the number is 2645.12, the vectors should be [2 6 4 5] and [1 2]. The whole number can be from 0 to 10 digits long and the decimal part 0 to 6. Check the validity of the function using a few numbers of your choice. A fenced enclosure consists of a rectangle of length L and width 2R, and a semicircle of radius R, as shown in Figure. The enclosure is to be built to have an area A of 1600 ft2. The cost of the fence is $40 per foot for the curved portion, and $30 per foot for the straight sides. Determine the values of R and L required to minimize the total cost of the fence and the minimum cost using calculus approach. A water tank consists of a cylindrical part of radius r and height h, and a hemispherical top. The tank is to be constructed to hold 500 meter3 of fluid when filled. The cost to construct the cylindrical part of the tank is $300 per square meter of the surface area; the hemispherical part costs $400 per square meter. Determine the radius that results in the least cost and compute the corresponding height and the cost using graphical approach. Verify your results using the calculus approach. A ceramic tile has the design shown in the figure. The shaded area is painted black and the rest of the tile is white. The border line between the red and the white areas follows the equation: y=Asin(x) Determine A such that the area of the white and black colors will be the same.

EGR 140 Scientific Programming Assignment # 7 Spring 2017 Use MATLAB to solve each problem by writing script files; copy and paste the script file AND the results in the Command Window and/or plot in the Figure Window to a WORD document that has your name and section in the headers of each page and the page number in each footer. Edit the output to remove extra lines and empty spaces. The script files SHOULD have comments for easy readability; take a print out of the Word file and staple before submission. Due by 3 PM on April 11, 2017. Write a used-defined function that calculates the average and the standard deviation of a list of numbers. Use the function to calculate the average and the standard deviation of the following list of grades: 80 75 91 60 79 89 65 80 95 50 81 Note: The average x_ave (or mean) of a given set of n number x_1,x_2,…..,x_n is given by: x_ave=(x_1+x_2+x_3+⋯+x_n)/n The standard deviation is given by: σ=√((∑_(i=1)^(i=n)▒(x_i-x_ave )^2 )/(n-1)) DO not use built-in functions to calculate the mean and the standard deviation. Write a user-defined function that arranges the digits of a given (positive) number in a row vector in the same order as they appear in the number; the function should also arrange the digits in the decimal part in a different vector. For example, if the number is 2645.12, the vectors should be [2 6 4 5] and [1 2]. The whole number can be from 0 to 10 digits long and the decimal part 0 to 6. Check the validity of the function using a few numbers of your choice. A fenced enclosure consists of a rectangle of length L and width 2R, and a semicircle of radius R, as shown in Figure. The enclosure is to be built to have an area A of 1600 ft2. The cost of the fence is $40 per foot for the curved portion, and $30 per foot for the straight sides. Determine the values of R and L required to minimize the total cost of the fence and the minimum cost using calculus approach. A water tank consists of a cylindrical part of radius r and height h, and a hemispherical top. The tank is to be constructed to hold 500 meter3 of fluid when filled. The cost to construct the cylindrical part of the tank is $300 per square meter of the surface area; the hemispherical part costs $400 per square meter. Determine the radius that results in the least cost and compute the corresponding height and the cost using graphical approach. Verify your results using the calculus approach. A ceramic tile has the design shown in the figure. The shaded area is painted black and the rest of the tile is white. The border line between the red and the white areas follows the equation: y=Asin(x) Determine A such that the area of the white and black colors will be the same.

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CEE 260 / MIE 273 Probability & Statistics Name: Final Exam, version D — 100 points (120 minutes) PLEASE READ QUESTIONS CAREFULLY and SHOW YOUR WORK! CALCULATORS PERMITTED – ABSOLUTELY NO REFERENCES! 1. Suppose the waiting time (in minutes) for your 911 SC Targa to reach operating temperature in the morning is uniformly distributed on [0,10], while the waiting time in the evening is uniformly distributed on [0,5] independent of morning waiting time. a. (5%) If you drive your Targa each morning and evening for a week (5 morning and 5 evening rides), what is the variance of your total waiting time? b. (5%) What is the expected value of the difference between morning and evening waiting time on a given day? 2. (10%) Find the maximum likelihood estimator (MLE) of ϴ when Xi ~ Exponential(ϴ) and you have observed X1, X2, X3, …, Xn. 2 3. The waiting time for delivery of a new Porsche 911 Carrera at the local dealership is distributed exponentially with a population mean of 3.55 months and population standard deviation of 1.1 months. Recently, in an effort to reduce the waiting time, the dealership has experimented with an online ordering system. A sample of 100 customers during a recent sales promotion generated a mean waiting time of 3.18 months using the new system. Assume that the population standard deviation of the waiting time has not changed from 1.1 months. (hint: the source distribution is irrelevant, but its parameters are relevant) a. (15%) What is the probability that the average wait time is between 3.2 and 6.4 months? (hint: draw a sketch for full credit) b. (10%) At the 0.05 level of significance, using the critical values approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? c. (10%) At the 0.01 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? 3 4. Porsche AG is a leading manufacturer of performance automobiles. The 911 Carrera model, Porsche’s premier sports car, reaches a top track speed of 180 miles per hour. Engineers claim the new advanced technology 911 GT2 automatically adjusts its top speed depending on the weather conditions. Suppose that in an effort to test this claim, Porsche selects a few 911 GT2 models to test drive on the company track in Stuttgart, Germany. The average top speed for the sample of 25 test drives is 182.36 mph, with a standard deviation of 7.24 mph. a. (5%) Without using complete sentences, what might be some problems with the sampling conducted above? Identify and explain at least 2. b. (15%) Using the critical values approach to hypothesis testing and a 0.10 level of significance, is there evidence that the mean top track speed is different for the 911 GT2? (hint: state the null and alternative hypotheses, draw a sketch, and show your work for full credit) c. (10%) Set up a 95% confidence interval estimate of the population mean top speed of the 911 GT2. d. (5%) Compare the results of (b) and (c). What conclusions do you reach about the top speed of the new 911 GT2? 4 5. (10%) Porsche USA believes that sales of the venerable 911 Carrera are a function of annual income (in thousands of dollars) and a risk tolerance index of the potential buyer. Determine the regression equation and provide a succinct analysis of Porsche’s conjecture using the following Excel results. SUMMARY OUTPUT Regression Stat istics Multiple R 0.805073 R Square 0.648142 Adjusted R Square 0.606747 Standard Error 7.76312 Observations 20 ANOVA df SS MS F Significance F Regression 2 1887.227445 943.6137225 15.65747206 0.000139355 Residual 17 1024.522555 60.26603265 Total 19 2911.75 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23.50557 6.845545641 3.433702952 0.003167982 9.062731576 37.94840898 Income 0.613408 0.125421229 4.890786567 0.000137795 0.348792801 0.878024121 Risk Index -0.00126 0.004519817 -0.278357691 0.784095184 -0.010794106 0.008277854 BONUS (5 points) What is the probability that 2 or more students in our class of 22 have the same birthday?

CEE 260 / MIE 273 Probability & Statistics Name: Final Exam, version D — 100 points (120 minutes) PLEASE READ QUESTIONS CAREFULLY and SHOW YOUR WORK! CALCULATORS PERMITTED – ABSOLUTELY NO REFERENCES! 1. Suppose the waiting time (in minutes) for your 911 SC Targa to reach operating temperature in the morning is uniformly distributed on [0,10], while the waiting time in the evening is uniformly distributed on [0,5] independent of morning waiting time. a. (5%) If you drive your Targa each morning and evening for a week (5 morning and 5 evening rides), what is the variance of your total waiting time? b. (5%) What is the expected value of the difference between morning and evening waiting time on a given day? 2. (10%) Find the maximum likelihood estimator (MLE) of ϴ when Xi ~ Exponential(ϴ) and you have observed X1, X2, X3, …, Xn. 2 3. The waiting time for delivery of a new Porsche 911 Carrera at the local dealership is distributed exponentially with a population mean of 3.55 months and population standard deviation of 1.1 months. Recently, in an effort to reduce the waiting time, the dealership has experimented with an online ordering system. A sample of 100 customers during a recent sales promotion generated a mean waiting time of 3.18 months using the new system. Assume that the population standard deviation of the waiting time has not changed from 1.1 months. (hint: the source distribution is irrelevant, but its parameters are relevant) a. (15%) What is the probability that the average wait time is between 3.2 and 6.4 months? (hint: draw a sketch for full credit) b. (10%) At the 0.05 level of significance, using the critical values approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? c. (10%) At the 0.01 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? 3 4. Porsche AG is a leading manufacturer of performance automobiles. The 911 Carrera model, Porsche’s premier sports car, reaches a top track speed of 180 miles per hour. Engineers claim the new advanced technology 911 GT2 automatically adjusts its top speed depending on the weather conditions. Suppose that in an effort to test this claim, Porsche selects a few 911 GT2 models to test drive on the company track in Stuttgart, Germany. The average top speed for the sample of 25 test drives is 182.36 mph, with a standard deviation of 7.24 mph. a. (5%) Without using complete sentences, what might be some problems with the sampling conducted above? Identify and explain at least 2. b. (15%) Using the critical values approach to hypothesis testing and a 0.10 level of significance, is there evidence that the mean top track speed is different for the 911 GT2? (hint: state the null and alternative hypotheses, draw a sketch, and show your work for full credit) c. (10%) Set up a 95% confidence interval estimate of the population mean top speed of the 911 GT2. d. (5%) Compare the results of (b) and (c). What conclusions do you reach about the top speed of the new 911 GT2? 4 5. (10%) Porsche USA believes that sales of the venerable 911 Carrera are a function of annual income (in thousands of dollars) and a risk tolerance index of the potential buyer. Determine the regression equation and provide a succinct analysis of Porsche’s conjecture using the following Excel results. SUMMARY OUTPUT Regression Stat istics Multiple R 0.805073 R Square 0.648142 Adjusted R Square 0.606747 Standard Error 7.76312 Observations 20 ANOVA df SS MS F Significance F Regression 2 1887.227445 943.6137225 15.65747206 0.000139355 Residual 17 1024.522555 60.26603265 Total 19 2911.75 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23.50557 6.845545641 3.433702952 0.003167982 9.062731576 37.94840898 Income 0.613408 0.125421229 4.890786567 0.000137795 0.348792801 0.878024121 Risk Index -0.00126 0.004519817 -0.278357691 0.784095184 -0.010794106 0.008277854 BONUS (5 points) What is the probability that 2 or more students in our class of 22 have the same birthday?

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AUCS 340: Ethics in the Professions Homework Assignment: International and US Health Care Systems The following homework assignment will help you to discover some of the differences between the administration of health care in the United States and internationally. This is a research based assignment; remember the use of Wikipedia.com is not an acceptable reference site for this course. You must include a references cited page for this assignment; correctly formatted APA or MLA references are acceptable (simply stating s web address is NOT a complete reference). The answers should be presented in paragraph formation. Staple all pages together for presentation. The first question refers to a country other than the United States of America 1) Socialized Medicine – provide a definition of the term socialized medicine and discuss a country that currently has a socialized medicine system to cover all citizens; this discussion should include the types of services offered to the citizens of this country. When was this system first implemented in this country? What is the name of this country’s health insurance plan? Compare the ranking for the life expectancy for this country to that of the United States. Which is higher? Why? Compare the cost of financing healthcare in this country to the United States in comparison to the amount of annual funding in dollars and the percentage of gross domestic product spent on health care for each country. What rank does this country have in comparison to the United States for overall health of its citizens? (This portion of the assignment should be approximately one page in length and graphic data is acceptable to support some answers, however, graphic information should only be used to explain your written explanation not as the answer to the question.) Bonus: Is this country’s system currently financially stable? Why or why not? The following questions refer to the delivery of healthcare in the United States of America, as it was organized prior to the implementation of the Affordable Care Act (ACA). The ACA is currently being phased into coverage. It is estimated that the answers to the following questions will result in an additional two to three pages of written text in addition to the page for question number one. 2) Medicare – when was it enacted? Who does it cover? Who was President when Medicare was originally passed? What do the specific portions Part A, Part B and Part D cover? When was Part D enacted? Who was President when Part D was enacted? Is the Medicare system currently financially stable? Why or why not. Compare the average life expectancy for males and females when Medicare was originally passed and the average life expectancy of males and females as of 2010; more recent data is acceptable. Bonus: What does Part C cover and when was it enacted? 3) Health Maintenance Organization (HMO) – Define the term health maintenance organization. When did this type of health insurance plan become popular in the United States? How does this type of system provide medical care to the people enrolled? This answer should discuss in network versus out of network coverage. 4) Medicaid- when was it enacted? Who does it cover? Who was President when this insurance plan was enacted? Are the coverage benefits the same state to state? Why or why not? Is the system currently financially stable? Why or why not. What effect does passage of the ACA project to have on enrollment in the Medicaid system? Why? 5) Organ Transplants – What is the mechanism for placement of a patient’s name on the organ transplant list? What is the current length of time a patient must wait for a heart transplant? Explain at least one reason why transplants are considered an ethical issue. How are transplants financed? Give at least one example of how much any type of organ transplant would cost. 6) Health Insurance/Information Portability and Accountability Act (HIPAA) – When was it enacted? Who was President when this legislation as passed? What is the scope of this legislative for the medical community and the general community? (Hint: There are actually two reasons for HIPAA legislation; make sure to state both in your response) 7) Death with Dignity Act – what year was the Oregon Death with Dignity Act passed? What ethical issue is covered by the Death with Dignity Act? List the factors that must be met for a patient to use the Death with Dignity Act. List two additional states that have enacted Death with Dignity Acts and when was the legislation passed in these states? 8) Hospice – what is hospice care? When was it developed? What country was most instrumental in the development of hospice care? Do health insurance plans in the United States cover hospice care? What types of services are covered for hospice care? Grading: 1) Accuracy and completeness of responses = 60% of grade 2) Correct use of sentence structure, spelling and grammar = 30% of grade 3) Appropriate use of references and citations = 10% of grade Simply stating a web page is not an appropriate reference This assignment is due on the date published in the course syllabus.

AUCS 340: Ethics in the Professions Homework Assignment: International and US Health Care Systems The following homework assignment will help you to discover some of the differences between the administration of health care in the United States and internationally. This is a research based assignment; remember the use of Wikipedia.com is not an acceptable reference site for this course. You must include a references cited page for this assignment; correctly formatted APA or MLA references are acceptable (simply stating s web address is NOT a complete reference). The answers should be presented in paragraph formation. Staple all pages together for presentation. The first question refers to a country other than the United States of America 1) Socialized Medicine – provide a definition of the term socialized medicine and discuss a country that currently has a socialized medicine system to cover all citizens; this discussion should include the types of services offered to the citizens of this country. When was this system first implemented in this country? What is the name of this country’s health insurance plan? Compare the ranking for the life expectancy for this country to that of the United States. Which is higher? Why? Compare the cost of financing healthcare in this country to the United States in comparison to the amount of annual funding in dollars and the percentage of gross domestic product spent on health care for each country. What rank does this country have in comparison to the United States for overall health of its citizens? (This portion of the assignment should be approximately one page in length and graphic data is acceptable to support some answers, however, graphic information should only be used to explain your written explanation not as the answer to the question.) Bonus: Is this country’s system currently financially stable? Why or why not? The following questions refer to the delivery of healthcare in the United States of America, as it was organized prior to the implementation of the Affordable Care Act (ACA). The ACA is currently being phased into coverage. It is estimated that the answers to the following questions will result in an additional two to three pages of written text in addition to the page for question number one. 2) Medicare – when was it enacted? Who does it cover? Who was President when Medicare was originally passed? What do the specific portions Part A, Part B and Part D cover? When was Part D enacted? Who was President when Part D was enacted? Is the Medicare system currently financially stable? Why or why not. Compare the average life expectancy for males and females when Medicare was originally passed and the average life expectancy of males and females as of 2010; more recent data is acceptable. Bonus: What does Part C cover and when was it enacted? 3) Health Maintenance Organization (HMO) – Define the term health maintenance organization. When did this type of health insurance plan become popular in the United States? How does this type of system provide medical care to the people enrolled? This answer should discuss in network versus out of network coverage. 4) Medicaid- when was it enacted? Who does it cover? Who was President when this insurance plan was enacted? Are the coverage benefits the same state to state? Why or why not? Is the system currently financially stable? Why or why not. What effect does passage of the ACA project to have on enrollment in the Medicaid system? Why? 5) Organ Transplants – What is the mechanism for placement of a patient’s name on the organ transplant list? What is the current length of time a patient must wait for a heart transplant? Explain at least one reason why transplants are considered an ethical issue. How are transplants financed? Give at least one example of how much any type of organ transplant would cost. 6) Health Insurance/Information Portability and Accountability Act (HIPAA) – When was it enacted? Who was President when this legislation as passed? What is the scope of this legislative for the medical community and the general community? (Hint: There are actually two reasons for HIPAA legislation; make sure to state both in your response) 7) Death with Dignity Act – what year was the Oregon Death with Dignity Act passed? What ethical issue is covered by the Death with Dignity Act? List the factors that must be met for a patient to use the Death with Dignity Act. List two additional states that have enacted Death with Dignity Acts and when was the legislation passed in these states? 8) Hospice – what is hospice care? When was it developed? What country was most instrumental in the development of hospice care? Do health insurance plans in the United States cover hospice care? What types of services are covered for hospice care? Grading: 1) Accuracy and completeness of responses = 60% of grade 2) Correct use of sentence structure, spelling and grammar = 30% of grade 3) Appropriate use of references and citations = 10% of grade Simply stating a web page is not an appropriate reference This assignment is due on the date published in the course syllabus.