## Determine the changes in enthalpy , internal energy and entropy when 2.7 kg of water taken at P1 = 1.0133 X 10^5 Pa and T1 = 293 K evaporate at P2 = 0.50665 X 10^5 Pa and T2 = 373 K . Cp(l) ~ Cv(l) = 4.187 X 10^3 J/kg/K the specific heat of evaporation being 2260.98 X 10^3 J/kg

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## Assignment 7 Due: 11:59pm on Friday, March 21, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 8.5 The figure shows two balls of equal mass moving in vertical circles. Part A Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal speed? ANSWER: Correct The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Part B Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal angular velocity? ANSWER: Correct A Mass on a Turntable: Conceptual A small metal cylinder rests on a circular turntable that is rotating at a constant rate, as illustrated in the diagram. Part A Which of the following sets of vectors best describes the velocity, acceleration, and net force acting on the cylinder at the point indicated in the diagram? The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Typesetting math: 100% Hint 1. The direction of acceleration can be determined from Newton’s second law According to Newton’s second law, the acceleration of an object has the same direction as the net force acting on that object. ANSWER: Correct Part B Let be the distance between the cylinder and the center of the turntable. Now assume that the cylinder is moved to a new location from the center of the turntable. Which of the following statements accurately describe the motion of the cylinder at the new location? Check all that apply. a b c d e R R/2 Typesetting math: 100% Hint 1. Find the speed of the cylinder Find the speed of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . ANSWER: Hint 2. Find the acceleration of the cylinder Find the magnitude of the acceleration of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . Hint 1. Centripetal acceleration Recall that the acceleration of an object that moves in a circular path of radius with constant speed has magnitude given by . Note that both the velocity and radius of the trajectory change when the cylinder is moved. ANSWER: ANSWER: v T R T v = R T a T R T r v a = v2 r a = 22R T 2 Typesetting math: 100% Correct Accelerating along a Racetrack A road race is taking place along the track shown in the figure . All of the cars are moving at constant speeds. The car at point F is traveling along a straight section of the track, whereas all the other cars are moving along curved segments of the track. Part A Let be the velocity of the car at point A. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path The speed of the cylinder has decreased. The speed of the cylinder has increased. The magnitude of the acceleration of the cylinder has decreased. The magnitude of the acceleration of the cylinder has increased. The speed and the acceleration of the cylinder have not changed. v A Typesetting math: 100% Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct Part B Let be the velocity of the car at point C. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: v v The acceleration is parallel to . The acceleration is perpendicular to and directed toward the inside of the track. The acceleration is perpendicular to and directed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v A v A v A v A v C v v Typesetting math: 100% Correct Part C Let be the velocity of the car at point D. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v C v C v C v C v D v v The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v D v D v D v D Typesetting math: 100% Part D Let be the velocity of the car at point F. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a straight path The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the object has a nonzero acceleration only if the magnitude of its velocity changes in time. ANSWER: Correct Part E Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has the acceleration of the least magnitude? Use A for the car at point A, B for the car at point B, and so on. Express your answer as the name the car that has the greatest magnitude of acceleration followed by the car with the least magnitude of accelation, and separate your answers with a comma. Hint 1. How to approach the problem Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is inversely proportional to the radius of curvature of the path. ANSWER: v F The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v F v F v F v F Typesetting math: 100% Correct Part F Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of car E. Hint 1. Find the acceleration of the car at point E Let be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude of the acceleration of the car at point E? Express your answer in terms of the radius of curvature and the speed of car E. Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: Hint 2. Find the acceleration of the car at point A If , what is the acceleration of the car at point A? Let be the radius of the two curves along which the cars at points A and E are traveling. Express your answer in terms of the speed of the car at E and the radius . r aE r vE a v r a = v2 r aE = vE 2 r vA = 2vE aA r vE r Typesetting math: 100% Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: ANSWER: Correct Problem 8.5 A 1300 car takes a 50- -radius unbanked curve at 13 . Part A What is the size of the friction force on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: v r a = v2 r aA = 4vE 2 r The magnitude of the acceleration of the car at point A is twice that of the car at point E. The magnitude of the acceleration of the car at point A is the same as that of the car at point E. The magnitude of the acceleration of the car at point A is half that of the car at point E. The magnitude of the acceleration of the car at point A is four times that of the car at point E. kg m m/s Typesetting math: 100% Correct Problem 8.10 It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1600- -diameter cylinder that rotates about its axis. The inside surface is the deck of the space station. Part A What rotation period will provide “normal” gravity? Express your answer with the appropriate units. ANSWER: Correct Problem 8.7 In the Bohr model of the hydrogen atom, an electron orbits a proton at a distance of . The proton pulls on the electron with an electric force of . Part A How many revolutions per second does the electron make? Express your answer with the appropriate units. ANSWER: fs = 4400 N m T = 56.8 s (mass m = 9.1 × 10−31 kg) 5.3 × 10−11 m 8.2 × 10−8 N Typesetting math: 100% Correct Problem 8.14 The weight of passengers on a roller coaster increases by 56 as the car goes through a dip with a 38 radius of curvature. Part A What is the car’s speed at the bottom of the dip? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 8.18 While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 14 , and you use your watch to find that each loop around takes 24 . Part A What is your speed? Express your answer to two significant figures and include the appropriate units. ANSWER: 6.56×1015 rev s % m v = 14 ms m s v = 3.7 ms Typesetting math: 100% Correct Part B What is the magnitude of your acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the ratio of your weight at the top of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: Correct Part D What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: a = 0.96 m s2 = 0.90 wtop FG Typesetting math: 100% Correct Enhanced EOC: Problem 8.46 A heavy ball with a weight of 120 is hung from the ceiling of a lecture hall on a 4.4- -long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.6 as it passes through the lowest point. You may want to review ( pages 201 – 204) . For help with math skills, you may want to review: Solutions of Systems of Equations Part A What is the tension in the rope at that point? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a free-body diagram indicating the forces acting on the ball when it is at its lowest point. Choose a coordinate system. What is the direction of the acceleration in your chosen coordinate system? What is the magnitude of the acceleration for the mass, which is moving in a circular path? What is Newton’s second law applied to the mass at the bottom of its swing? Make sure to use your coordinate system when determining the signs of all the forces and the acceleration. What is the tension in the rope at this point? ANSWER: = 1.1 wbottom FG N m m/s T = 210 N Typesetting math: 100% Correct Problem 8.43 In an amusement park ride called The Roundup, passengers stand inside a 16.0 -diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure . Part A Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the top of the ride? Express your answer with the appropriate units. ANSWER: Correct Part B m s kg 211 N Typesetting math: 100% Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the bottom of the ride? Express your answer with the appropriate units. ANSWER: Correct Part C What is the longest rotation period of the wheel that will prevent the riders from falling off at the top? Express your answer with the appropriate units. ANSWER: Correct Conceptual Question 9.9 A 2 object is moving to the right with a speed of 1 when it experiences an impulse of 6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: s kg 1270 N 5.68 s kg m/s i ^ Ns i ^ v = 4 ms Typesetting math: 100% Correct Part B What is the object’s direction after the impulse? ANSWER: Correct Conceptual Question 9.10 A 2 object is moving to the right with a speed of 2 when it experiences an impulse of -6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the impulse? to the right to the left kg m/s i ^ Ns i ^ v = 1 ms Typesetting math: 100% ANSWER: Correct Problem 9.5 Part A In the figure , what value of gives an impulse of 6.4 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct to the right to the left Fmax Ns Fmax = 1.6×103 N Typesetting math: 100% Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by . The impulse is a vector pointing in the same direction as the force vector. The units of are or . Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object’s momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object’s motion: . So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of . Part A Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: Correct We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure. (F J J = F) t J N * s kg * m/s p = mv )p = J = F) t N 0.70 × 10−3 s J J = 8.4 N * s Typesetting math: 100% Part B The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph in the figure. For this graph, the length height area slope of the rectangle corresponds to the impulse. Typesetting math: 100% Part C If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: maximum force area slope Typesetting math: 100% Correct When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as . Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem. Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction. ANSWER: F avg )t )t J = F )t avg J = )p = m(vf − vi ) Typesetting math: 100% Correct Part E Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball’s velocity just after leaving the bat if the bat applies an impulse of to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Problem 9.9 A 2.6 object is moving to the right with a speed of 1.0 when it experiences the force shown in the figure. The impulse on the ball caused by the bat will be in the positive negative x direction. kg m/s −8.4 N * s v = -26 m/s kg m/s Typesetting math: 100% Part A What is the object’s speed after the force ends? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the force ends? ANSWER: Correct Enhanced EOC: Problem 9.27 A tennis player swings her 1000 g racket with a speed of 11.0 . She hits a 60 g tennis ball that was approaching her at a speed of 19.0 . The ball rebounds at 41.0 . You may want to review ( pages 226 – 232) . For help with math skills, you may want to review: v = 0.62 ms to the right to the left m/s m/s m/s Typesetting math: 100% Solving Algebraic Equations Part A How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision. Express your answer with the appropriate units. Hint 1. How to approach the problem Given that you can ignore the interaction of the racket with her hand during the collision, what is conserved during the collision? Draw a picture indicating the direction of the racket and ball before the collision and a separate picture for after the collision. Place a coordinate system on your pictures, indicating the positive x direction. Keeping in mind that velocity can be either positive or negative in your coordinate system, what is the initial momentum of the ball–racket system? What is the final momentum of the ball–racket system in terms of the velocity of the racket after the collision? Using conservation of momentum, what are the velocity and speed of the racket after the collision? ANSWER: Correct Part B If the tennis ball and racket are in contact for 8.00 , what is the average force that the racket exerts on the ball? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the impulse on the ball related to the change in momentum of the ball? What is the change in momentum of the ball? How are the impulse on the ball and the collision time related to the average force on the ball? 7.40 ms ms Typesetting math: 100% ANSWER: Correct Problem 9.14 A 2.00×104 railroad car is rolling at 6.00 when a 6000 load of gravel is suddenly dropped in. Part A What is the car’s speed just after the gravel is loaded? Express your answer with the appropriate units. ANSWER: Correct Problem 9.17 A 330 bird flying along at 5.0 sees a 9.0 insect heading straight toward it with a speed of 34 (as measured by an observer on the ground, not by the bird). The bird opens its mouth wide and enjoys a nice lunch. Part A What is the bird’s speed immediately after swallowing? Express your answer to two significant figures and include the appropriate units. ANSWER: 450 N kg m/s kg 4.62 ms g m/s g m/s Typesetting math: 100% Correct Problem 9.20 A 50.0 archer, standing on frictionless ice, shoots a 200 arrow at a speed of 200 . Part A What is the recoil speed of the archer? Express your answer with the appropriate units. ANSWER: Correct Problem 9.25 A 40.0 ball of clay traveling east at 4.50 collides and sticks together with a 50.0 ball of clay traveling north at 4.50 . Part A What is the speed of the resulting ball of clay? Express your answer with the appropriate units. ANSWER: v = 4.0 ms kg g m/s 0.800 ms g m/s g m/s 3.20 ms Typesetting math: 100% Correct Problem 9.32 A particle of mass is at rest at . Its momentum for is given by , where is in . Part A Find an expression for , the force exerted on the particle as a function of time. Express your answer in terms of the given quantities. ANSWER: Correct Problem 9.37 Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 and a mass of 1.2×1013 hits the earth with an impact speed of 4.5×104 . Part A What is the earth’s recoil speed after such a collision? (Use a reference frame in which the earth was initially at rest.) Assume that . Express your answer to two significant figures and include the appropriate units. ANSWER: m t = 0 t > 0 px = 6t2 kgm/s t s Fx(t) Fx = 12t N km kg m/s MEarth= 5.98 × 1024 kg = 9.0×10−8 v ms Typesetting math: 100% Correct Part B What percentage is this of the earth’s speed around the sun? (Use the astronomical data in the textbook.) Express your answer using two significant figures. ANSWER: Correct Problem 9.42 One billiard ball is shot east at 1.8 . A second, identical billiard ball is shot west at 1.2 . The balls have a glancing collision, not a head-on collision, deflecting the second ball by 90 and sending it north at 1.50 . Part A What is the speed of the first ball after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the direction of the first ball after the collision? Give the direction as an angle south of east. = 3.0×10−10 of v % the earth’s speed m/s m/s 1 m/s v = 1.6 ms Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 9.49 Two 490 blocks of wood are 2.0 apart on a frictionless table. A 12 bullet is fired at 420 toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is 5.6 . Part A What is the speed of the second block after the bullet stops? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 99.5%. You received 156.21 out of a possible total of 157 points. = 68 1 g m g m/s m/s v = 4.6 ms Typesetting math: 100%

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## Assignment 8 Due: 11:59pm on Friday, April 4, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 10.3 Part A If a particle’s speed increases by a factor of 5, by what factor does its kinetic energy change? ANSWER: Correct Conceptual Question 10.11 A spring is compressed 1.5 . Part A How far must you compress a spring with twice the spring constant to store the same amount of energy? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct = 25 K2 K1 cm x = 1.1 cm Problem 10.2 The lowest point in Death Valley is below sea level. The summit of nearby Mt. Whitney has an elevation of 4420 . Part A What is the change in potential energy of an energetic 80 hiker who makes it from the floor of Death Valley to the top of Mt.Whitney? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 10.3 Part A At what speed does a 1800 compact car have the same kinetic energy as a 1.80×104 truck going 21.0 ? Express your answer with the appropriate units. ANSWER: Correct Problem 10.5 A boy reaches out of a window and tosses a ball straight up with a speed of 13 . The ball is 21 above the ground as he releases it. 85m m kg U = 3.5×106 J kg kg km/hr vc = 66.4 km hr m/s m Part A Use energy to find the ball’s maximum height above the ground. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B Use energy to find the ball’s speed as it passes the window on its way down. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Use energy to find the speed of impact on the ground. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Hmax = 30 m v = 13 ms v = 24 ms Problem 10.8 A 59.0 skateboarder wants to just make it to the upper edge of a “quarter pipe,” a track that is one-quarter of a circle with a radius of 2.30 . Part A What speed does he need at the bottom? Express your answer with the appropriate units. ANSWER: Correct Problem 10.12 A 1500 car traveling at 12 suddenly runs out of gas while approaching the valley shown in the figure. The alert driver immediately puts the car in neutral so that it will roll. Part A kg m 6.71 ms kg m/s What will be the car’s speed as it coasts into the gas station on the other side of the valley? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Ups and Downs Learning Goal: To apply the law of conservation of energy to an object launched upward in the gravitational field of the earth. In the absence of nonconservative forces such as friction and air resistance, the total mechanical energy in a closed system is conserved. This is one particular case of the law of conservation of energy. In this problem, you will apply the law of conservation of energy to different objects launched from the earth. The energy transformations that take place involve the object’s kinetic energy and its gravitational potential energy . The law of conservation of energy for such cases implies that the sum of the object’s kinetic energy and potential energy does not change with time. This idea can be expressed by the equation , where “i” denotes the “initial” moment and “f” denotes the “final” moment. Since any two moments will work, the choice of the moments to consider is, technically, up to you. That choice, though, is usually suggested by the question posed in the problem. First, let us consider an object launched vertically upward with an initial speed . Neglect air resistance. Part A As the projectile goes upward, what energy changes take place? ANSWER: v = 6.8 ms K = (1/2)mv2 U = mgh Ki + Ui = Kf + Uf v Correct Part B At the top point of the flight, what can be said about the projectile’s kinetic and potential energy? ANSWER: Correct Strictly speaking, it is not the ball that possesses potential energy; rather, it is the system “Earth-ball.” Although we will often talk about “the gravitational potential energy of an elevated object,” it is useful to keep in mind that the energy, in fact, is associated with the interactions between the earth and the elevated object. Part C The potential energy of the object at the moment of launch __________. ANSWER: Both kinetic and potential energy decrease. Both kinetic and potential energy increase. Kinetic energy decreases; potential energy increases. Kinetic energy increases; potential energy decreases. Both kinetic and potential energy are at their maximum values. Both kinetic and potential energy are at their minimum values. Kinetic energy is at a maximum; potential energy is at a minimum. Kinetic energy is at a minimum; potential energy is at a maximum. Correct Usually, the zero level is chosen so as to make the relevant calculations simpler. In this case, it makes good sense to assume that at the ground level–but this is not, by any means, the only choice! Part D Using conservation of energy, find the maximum height to which the object will rise. Express your answer in terms of and the magnitude of the acceleration of gravity . ANSWER: Correct You may remember this result from kinematics. It is comforting to know that our new approach yields the same answer. Part E At what height above the ground does the projectile have a speed of ? Express your answer in terms of and the magnitude of the acceleration of gravity . ANSWER: is negative is positive is zero depends on the choice of the “zero level” of potential energy U = 0 hmax v g hmax = v2 2g h 0.5v v g h = 3 v2 8g Correct Part F What is the speed of the object at the height of ? Express your answer in terms of and . Use three significant figures in the numeric coefficient. Hint 1. How to approach the problem You are being asked for the speed at half of the maximum height. You know that at the initial height ( ), the speed is . All of the energy is kinetic energy, and so, the total energy is . At the maximum height, all of the energy is potential energy. Since the gravitational potential energy is proportional to , half of the initial kinetic energy must have been converted to potential energy when the projectile is at . Thus, the kinetic energy must be half of its original value (i.e., when ). You need to determine the speed, as a multiple of , that corresponds to such a kinetic energy. ANSWER: Correct Let us now consider objects launched at an angle. For such situations, using conservation of energy leads to a quicker solution than can be produced by kinematics. Part G A ball is launched as a projectile with initial speed at an angle above the horizontal. Using conservation of energy, find the maximum height of the ball’s flight. Express your answer in terms of , , and . Hint 1. Find the final kinetic energy Find the final kinetic energy of the ball. Here, the best choice of “final” moment is the point at which the ball reaches its maximum height, since this is the point we are interested in. u (1/2)hmax v g h = 0 v (1/2)mv2 h (1/2)hmax (1/4)mv2 h = (1/2)hmax v u = 0.707v v hmax v g Kf Express your answer in terms of , , and . Hint 1. Find the speed at the maximum height The speed of the ball at the maximum height is __________. ANSWER: ANSWER: ANSWER: Correct Part H A ball is launched with initial speed from ground level up a frictionless slope. The slope makes an angle with the horizontal. Using conservation of energy, find the maximum vertical height to which the ball will climb. Express your answer in terms of , , and . You may or may not use all of these quantities. v m 0 v v cos v sin v tan Kf = 0.5m(vcos())2 hmax = (vsin())2 2g v hmax v g ANSWER: Correct Interestingly, the answer does not depend on . The difference between this situation and the projectile case is that the ball moving up a slope has no kinetic energy at the top of its trajectory whereas the projectile launched at an angle does. Part I A ball is launched with initial speed from the ground level up a frictionless hill. The hill becomes steeper as the ball slides up; however, the ball remains in contact with the hill at all times. Using conservation of energy, find the maximum vertical height to which the ball will climb. Express your answer in terms of and . ANSWER: Correct The profile of the hill does not matter; the equation would have the same terms regardless of the steepness of the hill. Problem 10.14 A 12- -long spring is attached to the ceiling. When a 2.2 mass is hung from it, the spring stretches to a length of 17 . Part A What is the spring constant ? Express your answer to two significant figures and include the appropriate units. hmax = v2 2g v hmax v g hmax = v2 2g Ki + Ui = Kf + Uf cm kg cm k ANSWER: Correct Part B How long is the spring when a 3.0 mass is suspended from it? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 10.17 A 6.2 mass hanging from a spring scale is slowly lowered onto a vertical spring, as shown in . You may want to review ( pages 255 – 257) . For help with math skills, you may want to review: Solving Algebraic Equations = 430 k Nm kg y = 19 cm kg Part A What does the spring scale read just before the mass touches the lower spring? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces acting on the mass before it touches the scale. What is the net force on the mass? What is the force on the mass due to gravity? What is the force on the mass due to the scale? ANSWER: Correct Part B The scale reads 22 when the lower spring has been compressed by 2.7 . What is the value of the spring constant for the lower spring? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces acting on the mass. What is the net force on the mass? What is the force on the mass due to gravity? What is the force on the mass due to the scale? Use these to determine the force on the mass by the spring, taking note of the directions from your picture. How is the spring constant related to the force by the spring and the compression of the spring? Check your units. ANSWER: F = 61 N N cm k = 1400 k Nm Correct Part C At what compression length will the scale read zero? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces on the mass. When the scale reads zero, what is the force on the mass due to the scale? What is the gravitational force on the mass? What is the force on the mass by the spring? How is the compression length related to the force by the spring and the spring constant? Check your units. ANSWER: Correct Problem 10.18 Part A How far must you stretch a spring with = 800 to store 180 of energy? Express your answer to two significant figures and include the appropriate units. ANSWER: y = 4.2 cm k N/m J Correct Problem 10.22 A 15 runaway grocery cart runs into a spring with spring constant 230 and compresses it by 57 . Part A What was the speed of the cart just before it hit the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Spring Gun A spring-loaded toy gun is used to shoot a ball straight up in the air. The ball reaches a maximum height , measured from the equilibrium position of the spring. s = 0.67 m kg N/m cm v = 2.2 ms H Part A The same ball is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the ball go this time? Neglect friction. Assume that the spring is ideal and that the distance by which the spring is compressed is negligible compared to . Hint 1. Potential energy of the spring The potential energy of a spring is proportional to the square of the distance the spring is compressed. The spring was compressed half the distance, so the mass, when launched, has one quarter of the energy as in the first trial. Hint 2. Potential energy of the ball At the highest point in the ball’s trajectory, all of the spring’s potential energy has been converted into gravitational potential energy of the ball. ANSWER: Correct A Bullet Is Fired into a Wooden Block A bullet of mass is fired horizontally with speed at a wooden block of mass resting on a frictionless table. The bullet hits the block and becomes completely embedded within it. After the bullet has come to rest within the block, the block, with the bullet in it, is traveling at speed . H height = H 4 mb vi mw vf Part A Which of the following best describes this collision? Hint 1. Types of collisions An inelastic collision is a collision in which kinetic energy is not conserved. In a partially inelastic collision, kinetic energy is lost, but the objects colliding do not stick together. From this information, you can infer what completely inelastic and elastic collisions are. ANSWER: Correct Part B Which of the following quantities, if any, are conserved during this collision? Hint 1. When is kinetic energy conserved? Kinetic energy is conserved only in perfectly elastic collisions. ANSWER: perfectly elastic partially inelastic perfectly inelastic Correct Part C What is the speed of the block/bullet system after the collision? Express your answer in terms of , , and . Hint 1. Find the momentum after the collision What is the total momentum of the block/bullet system after the collision? Express your answer in terms of and other given quantities. ANSWER: Hint 2. Use conservation of momentum The momentum of the block/bullet system is conserved. Therefore, the momentum before the collision is the same as the momentum after the collision. Find a second expression for , this time expressed as the total momentum of the system before the collision. Express your answer in terms of and other given quantities. ANSWER: kinetic energy only momentum only kinetic energy and momentum neither momentum nor kinetic energy vi mw mb ptotal vf ptotal = (mw + mb)vf ptotal vi ptotal = mbvi ANSWER: Correct Problem 10.31 Ball 1, with a mass of 150 and traveling at 15.0 , collides head on with ball 2, which has a mass of 340 and is initially at rest. Part A What are the final velocities of each ball if the collision is perfectly elastic? Express your answer with the appropriate units. ANSWER: Correct Part B Express your answer with the appropriate units. ANSWER: Correct Part C vf = mb vi mb+mw g m/s g (vfx) = -5.82 1 ms (vfx) = 9.18 2 ms What are the final velocities of each ball if the collision is perfectly inelastic? Express your answer with the appropriate units. ANSWER: Correct Part D Express your answer with the appropriate units. ANSWER: Correct Enhanced EOC: Problem 10.43 A package of mass is released from rest at a warehouse loading dock and slides down the = 2.2 – high, frictionless chute to a waiting truck. Unfortunately, the truck driver went on a break without having removed the previous package, of mass , from the bottom of the chute. You may want to review ( pages 265 – 269) . For help with math skills, you may want to review: Solving Algebraic Equations (vfx) = 4.59 1 ms (vfx) = 4.59 2 ms m h m 2m Part A Suppose the packages stick together. What is their common speed after the collision? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem There are two parts to this problem: the block sliding down the frictionless incline and the collision. What conservation laws are valid in each part? In terms of , what are the kinetic and potential energies of the block at the top of the incline? What is the potential energy of the same block at the bottom just before the collision? What are the kinetic energy and velocity of block just before the collision? What is conserved during the collision? What is the total momentum of the two blocks before the collision? What is the momentum of the two blocks stuck together after the collision? What is the velocity of the two blocks after the collision? ANSWER: Correct Part B Suppose the collision between the packages is perfectly elastic. To what height does the package of mass rebound? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem There are three parts to this problem: the block sliding down the incline, the collision, and mass going back up the incline. What conservation laws are valid in each part? m m v = 2.2 ms m m What is an elastic collision? For an elastic collision, how are the initial and final velocities related when one of the masses is initially at rest? Using the velocity of just before the collision from Part A, what is the velocity of just after the collision in this case? What are the kinetic and potential energies of mass just after the collision? What is the kinetic energy of mass at its maximum rebound height? Using conservation of energy, what is the potential energy of mass at its maximum height? What is the maximum height? ANSWER: Correct Problem 10.35 A cannon tilted up at a 35.0 angle fires a cannon ball at 79.0 from atop a 21.0 -high fortress wall. Part A What is the ball’s impact speed on the ground below? Express your answer with the appropriate units. ANSWER: Correct Problem 10.45 A 1000 safe is 2.5 above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it 48 . m m m m m h = 24 cm $ m/s m vf = 81.6 ms kg m cm Part A What is the spring constant of the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 10.49 A 100 block on a frictionless table is firmly attached to one end of a spring with = 21 . The other end of the spring is anchored to the wall. A 30 ball is thrown horizontally toward the block with a speed of 6.0 . Part A If the collision is perfectly elastic, what is the ball’s speed immediately after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the maximum compression of the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: = 2.5×105 k Nm g k N/m g m/s v = 3.2 ms Correct Part C Repeat part A for the case of a perfectly inelastic collision. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Repeat part B for the case of a perfectly inelastic collision. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 99.4%. You received 120.28 out of a possible total of 121 points. x = 0.19 m v = 1.4 ms x = 0.11 m

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## Chapter 13 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 Matter of Some Gravity Learning Goal: To understand Newton’s law of gravitation and the distinction between inertial and gravitational masses. In this problem, you will practice using Newton’s law of gravitation. According to that law, the magnitude of the gravitational force between two small particles of masses and , separated by a distance , is given by , where is the universal gravitational constant, whose numerical value (in SI units) is . This formula applies not only to small particles, but also to spherical objects. In fact, the gravitational force between two uniform spheres is the same as if we concentrated all the mass of each sphere at its center. Thus, by modeling the Earth and the Moon as uniform spheres, you can use the particle approximation when calculating the force of gravity between them. Be careful in using Newton’s law to choose the correct value for . To calculate the force of gravitational attraction between two uniform spheres, the distance in the equation for Newton’s law of gravitation is the distance between the centers of the spheres. For instance, if a small object such as an elephant is located on the surface of the Earth, the radius of the Earth would be used in the equation. Note that the force of gravity acting on an object located near the surface of a planet is often called weight. Also note that in situations involving satellites, you are often given the altitude of the satellite, that is, the distance from the satellite to the surface of the planet; this is not the distance to be used in the formula for the law of gravitation. There is a potentially confusing issue involving mass. Mass is defined as a measure of an object’s inertia, that is, its ability to resist acceleration. Newton’s second law demonstrates the relationship between mass, acceleration, and the net force acting on an object: . We can now refer to this measure of inertia more precisely as the inertial mass. On the other hand, the masses of the particles that appear in the expression for the law of gravity seem to have nothing to do with inertia: Rather, they serve as a measure of the strength of gravitational interactions. It would be reasonable to call such a property gravitational mass. Does this mean that every object has two different masses? Generally speaking, yes. However, the good news is that according to the latest, highly precise, measurements, the inertial and the gravitational mass of an object are, in fact, equal to each other; it is an established consensus among physicists that there is only one mass after all, which is a measure of both the object’s inertia and its ability to engage in gravitational interactions. Note that this consensus, like everything else in science, is open to possible amendments in the future. In this problem, you will answer several questions that require the use of Newton’s law of gravitation. Part A Two particles are separated by a certain distance. The force of gravitational interaction between them is . Now the separation between the particles is tripled. Find the new force of gravitational Fg m1 m2 r Fg = G m1m2 r2 G 6.67 × 10−11 Nm2 kg2 r r rEarth F = m net a F0 interaction . Express your answer in terms of . ANSWER: Part B A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite moves to a different orbit, so that its altitude is tripled. Find the new force of gravitational interaction . Express your answer in terms of . You did not open hints for this part. ANSWER: Part C A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite is brought back to the surface of the planet. Find the new force of gravitational interaction . Express your answer in terms of . ANSWER: F1 F0 F1 = F0 F2 F0 F2 = F0 F4 F0 Typesetting math: 81% Part D Two satellites revolve around the Earth. Satellite A has mass and has an orbit of radius . Satellite B has mass and an orbit of unknown radius . The forces of gravitational attraction between each satellite and the Earth is the same. Find . Express your answer in terms of . ANSWER: Part E An adult elephant has a mass of about 5.0 tons. An adult elephant shrew has a mass of about 50 grams. How far from the center of the Earth should an elephant be placed so that its weight equals that of the elephant shrew on the surface of the Earth? The radius of the Earth is 6400 . ( .) Express your answer in kilometers. ANSWER: The table below gives the masses of the Earth, the Moon, and the Sun. Name Mass (kg) Earth Moon Sun F4 = m r 6m rb rb r rb = r km 1 ton = 103 kg r = km 5.97 × 1024 7.35 × 1022 1.99 × 1030 Typesetting math: 81% The average distance between the Earth and the Moon is . The average distance between the Earth and the Sun is . Use this information to answer the following questions. Part F Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the new moon (when the moon is located directly between the Earth and the Sun). Express your answer in newtons to three significant figures. You did not open hints for this part. ANSWER: Part G Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the full moon (when the Earth is located directly between the moon and the sun). Express your answer in newtons to three significant figures. ANSWER: ± Understanding Newton’s Law of Universal Gravitation Learning Goal: To understand Newton’s law of universal gravitation and be able to apply it in two-object situations and (collinear) three-object situations; to distinguish between the use of and . 3.84 × 108 m 1.50 × 1011 m Fnet Fnet = N Fnet Fnet = N Typesetting math: 81% G g In the late 1600s, Isaac Newton proposed a rule to quantify the attractive force known as gravity between objects that have mass, such as those shown in the figure. Newton’s law of universal gravitation describes the magnitude of the attractive gravitational force between two objects with masses and as , where is the distance between the centers of the two objects and is the gravitational constant. The gravitational force is attractive, so in the figure it pulls to the right on (toward ) and toward the left on (toward ). The gravitational force acting on is equal in size to, but exactly opposite in direction from, the gravitational force acting on , as required by Newton’s third law. The magnitude of both forces is calculated with the equation given above. The gravitational constant has the value and should not be confused with the magnitude of the gravitational free-fall acceleration constant, denoted by , which equals 9.80 near the surface of the earth. The size of in SI units is tiny. This means that gravitational forces are sizeable only in the vicinity of very massive objects, such as the earth. You are in fact gravitationally attracted toward all the objects around you, such as the computer you are using, but the size of that force is too small to be noticed without extremely sensitive equipment. Consider the earth following its nearly circular orbit (dashed curve) about the sun. The earth has mass and the sun has mass . They are separated, center to center, by . Part A What is the size of the gravitational force acting on the earth due to the sun? Express your answer in newtons. F g m1 m2 Fg = G( ) m1m2 r2 r G m1 m2 m2 m1 m1 m2 G G = 6.67 × 10−11 N m2/kg2 g m/s2 G mearth = 5.98 × 1024 kg msun = 1.99 × 1030 kg r = 93 million miles = 150 million km Typesetting math: 81% You did not open hints for this part. ANSWER: 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). Part E This question will be shown after you complete previous question(s). Part F N Typesetting math: 81% This question will be shown after you complete previous question(s). Understanding Mass and Weight Learning Goal: To understand the distinction between mass and weight and to be able to calculate the weight of an object from its mass and Newton’s law of gravitation. The concepts of mass and weight are often confused. In fact, in everyday conversations, the word “weight” often replaces “mass,” as in “My weight is seventy-five kilograms” or “I need to lose some weight.” Of course, mass and weight are related; however, they are also very different. Mass, as you recall, is a measure of an object’s inertia (ability to resist acceleration). Newton’s 2nd law demonstrates the relationship among an object’s mass, its acceleration, and the net force acting on it: . Mass is an intrinsic property of an object and is independent of the object’s location. Weight, in contrast, is defined as the force due to gravity acting on the object. That force depends on the strength of the gravitational field of the planet: , where is the weight of an object, is the mass of that object, and is the local acceleration due to gravity (in other words, the strength of the gravitational field at the location of the object). Weight, unlike mass, is not an intrinsic property of the object; it is determined by both the object and its location. Part A Which of the following quantities represent mass? Check all that apply. ANSWER: Fnet = ma w = mg w m g 12.0 lbs 0.34 g 120 kg 1600 kN 0.34 m 411 cm 899 MN Typesetting math: 81% Part B This question will be shown after you complete previous question(s). Using the universal law of gravity, we can find the weight of an object feeling the gravitational pull of a nearby planet. We can write an expression , where is the weight of the object, is the gravitational constant, is the mass of that object, is mass of the planet, and is the distance from the center of the planet to the object. If the object is on the surface of the planet, is simply the radius of the planet. Part C The gravitational field on the surface of the earth is stronger than that on the surface of the moon. If a rock is transported from the moon to the earth, which properties of the rock change? ANSWER: Part D This question will be shown after you complete previous question(s). Part E If acceleration due to gravity on the earth is , which formula gives the acceleration due to gravity on Loput? You did not open hints for this part. ANSWER: w = GmM/r2 w G m M r r mass only weight only both mass and weight neither mass nor weight g Typesetting math: 81% 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 This question will be shown after you complete previous question(s). ± Weight on a Neutron Star Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter. g 1.7 5.6 g 1.72 5.6 g 1.72 5.62 g 5.6 1.7 g 5.62 1.72 g 5.6 1.72 Typesetting math: 81% Part A If you weigh 655 on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 19.0 ? Take the mass of the sun to be = 1.99×1030 , the gravitational constant to be = 6.67×10−11 , and the acceleration due to gravity at the earth’s surface to be = 9.810 . Express your weight in newtons. You did not open hints for this part. ANSWER: ± Escape Velocity Learning Goal: To introduce you to the concept of escape velocity for a rocket. The escape velocity is defined to be the minimum speed with which an object of mass must move to escape from the gravitational attraction of a much larger body, such as a planet of total mass . The escape velocity is a function of the distance of the object from the center of the planet , but unless otherwise specified this distance is taken to be the radius of the planet because it addresses the question “How fast does my rocket have to go to escape from the surface of the planet?” Part A The key to making a concise mathematical definition of escape velocity is to consider the energy. If an object is launched at its escape velocity, what is the total mechanical energy of the object at a very large (i.e., infinite) distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very large distances. You did not open hints for this part. ANSWER: N km ms kg G N m2/kg2 g m/s2 wstar wstar = N m M R Etotal Typesetting math: 81% Consider the motion of an object between a point close to the planet and a point very very far from the planet. Indicate whether the following statements are true or false. Part B Angular momentum about the center of the planet is conserved. ANSWER: Part C Total mechanical energy is conserved. ANSWER: Part D Kinetic energy is conserved. ANSWER: Etotal = true false true false Typesetting math: 81% 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). A Satellite in a Circular Orbit Consider a satellite of mass that orbits a planet of mass in a circle a distance from the center of the planet. The satellite’s mass is negligible compared with that of the planet. Indicate whether each of the statements in this problem is true or false. Part A The information given is sufficient to uniquely specify the speed, potential energy, and angular momentum of the satellite. You did not open hints for this part. ANSWER: true false m1 m2 r true false Typesetting math: 81% Part B The total mechanical energy of the satellite is conserved. You did not open hints for this part. ANSWER: Part C The linear momentum vector of the satellite is conserved. You did not open hints for this part. ANSWER: Part D The angular momentum of the satellite about the center of the planet is conserved. You did not open hints for this part. ANSWER: true false true false Typesetting math: 81% Part E The equations that express the conservation laws of total mechanical energy and linear momentum are sufficient to solve for the speed necessary to maintain a circular orbit at without using . You did not open hints for this part. ANSWER: At the Galaxy’s Core Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light years and an orbital speed of about 200 . Part A Determine the mass of the massive object at the center of the Milky Way galaxy. Take the distance of one light year to be . Express your answer in kilograms. You did not open hints for this part. true false R F = ma true false km/s M 9.461 × 1015 m Typesetting math: 81% ANSWER: 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). Part E This question will be shown after you complete previous question(s). Properties of Circular Orbits Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. M = kg Typesetting math: 81% The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit–a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass . For all parts of this problem, where appropriate, use for the universal gravitational constant. Part A Find the orbital speed for a satellite in a circular orbit of radius . Express the orbital speed in terms of , , and . You did not open hints for this part. ANSWER: Part B Find the kinetic energy of a satellite with mass in a circular orbit with radius . Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. ANSWER: Part C M G v R G M R v = K m R \texttip{K}{K} = Typesetting math: 81% This question will be shown after you complete previous question(s). Part D Find the orbital period \texttip{T}{T}. Express your answer in terms of \texttip{G}{G}, \texttip{M}{M}, \texttip{R}{R}, and \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Part E This question will be shown after you complete previous question(s). Part F Find \texttip{L}{L}, the magnitude of the angular momentum of the satellite with respect to the center of the planet. Express your answer in terms of \texttip{m}{m}, \texttip{M}{M}, \texttip{G}{G}, and \texttip{R}{R}. You did not open hints for this part. ANSWER: \texttip{T}{T} = Typesetting math: 81% Part G The quantities \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L} all represent physical quantities characterizing the orbit that depend on radius \texttip{R}{R}. Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answer as a comma-separated list of exponents corresponding to \texttip{v}{v}, \texttip{K}{K}, \texttip{U}{U}, and \texttip{L}{L}, in that order. For example, -1,-1/2,-0.5,-3/2 would mean v \propto R^{-1}, K \propto R^{-1/2}, and so forth. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \texttip{L}{L} = Typesetting math: 81%

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## Assignment 1 Due: 11:59pm on Wednesday, February 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 1.6 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Positive Negative Negative Positive Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Conceptual Question 1.7 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Positive Negative Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Enhanced EOC: Problem 1.18 The figure shows the motion diagram of a drag racer. The camera took one frame every 2 . Positive Negative Positive Negative Negative Positive s You may want to review ( pages 16 – 19) . For help with math skills, you may want to review: Plotting Points on a Graph Part A Make a position-versus-time graph for the drag racer. Hint 1. How to approach the problem Based on Table 1.1 in the book/e-text, what two observables are associated with each point? Which position or point of the drag racer occurs first? Which position occurs last? If you label the first point as happening at , at what time does the next point occur? At what time does the last position point occur? What is the position of a point halfway in between and ? Can you think of a way to estimate the positions of the points using a ruler? ANSWER: t = 0 s x = 0 m x = 200 m Correct Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Part A At what time(s) do the rockets have the same velocity? Hint 1. How to determine the velocity The diagram shows position, not velocity. You can’t find instantaneous velocity from this diagram, but you can determine the average velocity between two times and : . Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: Correct t1 t2 vavg[t1, t2] = x(t2)−x(t1) t2−t1 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Part B At what time(s) do the rockets have the same x position? ANSWER: Correct Part C At what time(s) do the two rockets have the same acceleration? Hint 1. How to determine the acceleration The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next. ANSWER: at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part F At what time(s) is rocket A ahead of rocket B? and nonzero acceleration velocity displacement time and nonzero acceleration velocity displacement time Hint 1. Use the diagram You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: Correct Dimensions of Physical Quantities Learning Goal: To introduce the idea of physical dimensions and to learn how to find them. Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and inches are different units they have the same dimension–length. Part A In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint 1. MKS system The current system of units is called the International System (abbreviated SI from the French Système International). In the past this system was called the mks system for its base units: meter, kilogram, and second. What are the dimensions of these quantities? ANSWER: before only after only before and after between and at no time(s) shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these three dimensions suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are called derived dimensions and the units they are measured in are called derived units. For example, area has derived dimensions . (Note that “dimensions of variable ” is symbolized as .) You can find these dimensions by looking at the formula for the area of a square , where is the length of a side of the square. Clearly . Plugging this into the equation gives . Part B Find the dimensions of volume. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for volume You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will give you the dimensions for volume. You can find the dimensions most easily from the volume of a cube , where is the length of the edge of the cube. ANSWER: acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge l m t A [A] = l2 x [x] A = s2 s [s] = l [A] = [s] = 2 l2 [V ] l m t V = e3 e [V ] = l3 Correct Part C Find the dimensions of speed. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for speed Speed is defined in terms of distance and time as . Therefore, . Hint 2. Familiar units for speed You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have the dimensions for speed. ANSWER: Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This means that , which comes from subtracting two speeds, has the same dimensions as speed. It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect. This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, is not a valid dimension for a [v] l m t v d t v = d t [v] = [d]/[t] [v] = lt−1 v l + t physical quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. ). Part D Find the dimensions of acceleration. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for acceleration In physics, acceleration is defined as the change in velocity in a certain time. This is shown by the equation . The is a symbol that means “the change in.” ANSWER: Correct Consistency of Units In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics. Part A Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation , where is the magnitude of the gravitational attraction on either body, and are the masses of the bodies, is the distance between them, and is the gravitational constant. In SI units, the units of force are , the units of mass are , and the units of distance are . For this equation to have consistent units, the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation m2/3 l2 t−2 [a] l m t a a = v/t [a] = lt−2 F = Gm1m2 r2 F m1 m2 r G kg m/s2 kg m G . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: Correct Part B One consequence of Einstein’s theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations: , where is mass, is the speed of the light, and is the energy. In SI units, the units of speed are . For the preceding equation to have consistent units (the same units on both sides of the equation), the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: F = Gm1m2 r2 m1 kg G kg3 ms2 kgs2 m3 m3 kgs2 m kgs2 E = mc2 m c E m/s E E = mc2 m kg E Correct To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for the units of the unknown variable. Problem 1.24 Convert the following to SI units: Part A 5.0 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B 54 Express your answer to two significant figures and include the appropriate units. kgm s kgm2 s2 kgs2 m2 kgm2 s m kg in 0.13 m ft/s ANSWER: Correct Part C 72 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D 17 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 1.55 The figure shows a motion diagram of a car traveling down a street. The camera took one frame every 10 . A distance scale is provided. 16 ms mph 32 ms in2 1.1×10−2 m2 s Part A Make a position-versus-time graph for the car. ANSWER: Incorrect; Try Again ± Moving at the Speed of Light Part A How many nanoseconds does it take light to travel a distance of 4.40 in vacuum? Express your answer numerically in nanoseconds. Hint 1. How to approach the problem Light travels at a constant speed; therefore, you can use the formula for the distance traveled in a certain amount of time by an object moving at constant speed. Before performing any calculations, it is often recommended, although it is not strictly necessary, to convert all quantities to their fundamental units rather than to multiples of the fundamental unit. km Hint 2. Find how many seconds it takes light to travel the given distance Given that the speed of light in vacuum is , how many seconds does it take light to travel a distance of 4.40 ? Express your answer numerically in seconds. Hint 1. Find the time it takes light to travel a certain distance How long does it take light to travel a distance ? Let be the speed of light. Hint 1. The speed of an object The equation that relates the distance traveled by an object with constant speed in a time is . ANSWER: Correct Hint 2. Convert the given distance to meters Convert = 4.40 to meters. Express your answer numerically in meters. Hint 1. Conversion of kilometers to meters Recall that . 3.00 × 108 m/s km r c s v t s = vt r c r c c r d km 1 km = 103 m ANSWER: Correct ANSWER: Correct Now convert the time into nanoseconds. Recall that . ANSWER: Correct Score Summary: Your score on this assignment is 84.7%. You received 50.84 out of a possible total of 60 points. 4.40km = 4400 m 1.47×10−5 s 1 ns = 10−9 s 1.47×104 ns

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## The value of DH° for the reaction below is -482 kJ. Calculate the heat (kJ) released to the surroundings when 24.0 g of CO (g) reacts completely. 2 2 2CO(g) +O (g)®2CO (g) A) 3 2.89×10 B) 103 C) 207 D) 65.7 E) -482

C) 207

## An average sample of coal contains 3.0% sulfur by mass. Calculate the moles of sulfur present in 2.40 103 kg of coal. (Points : 4) 9.4 102 mol 2.2 103 mol 6.0 103 mol 7.2 104 mol 7.5 104 mol

An average sample of coal contains 3.0% sulfur by mass. … Read More...

## Physics 2010 Sid Rudolph Fall 2009 MIDTERM 4 REVIEW Problems marked with an asterisk (*) are for the final. Solutions are on the course web page. 1. A. The drawing to the right shows glass tubing, a rubber bulb and two bottles. Is the situation you see possible? If so, carefully describe what has taken place in order to produce the situation depicted. B. The picture depicts three glass vessels, each filled with a liquid. The liquids each have different densities, and DA > DB > DC. In vessel B sits an unknown block halfway to the bottom and completely submerged. 1. _______ In which vessel would the block sit on the bottom? 2. _______ In which vessel would the block float on the top? 3. _______ In which vessel would the block feel the smallest buoyant force? 4. _______ In which vessels are buoyant forces on the block are the same? 5. _______ Assume the coefficient of volume expansion for the liquid in B and the block are $B > $block. If the temperature of vessel B with the block is raised, would block B rise to the surface, sink to the bottom, or remain where it is? 2. A circular tank with a 1.50 m radius is filled with two fluids, a 4.00 m layer of water and a 3.00 m layer of oil. Use Doil = 8.24 × 10 kg/m and Dwater = 1.00 × 10 kg/m , and Datm = 1.01 × 10 N/m . 2 3 3 3 5 2 A. What are the gauge and absolute pressures 1.00 m above the bottom of the tank? B. A block of material in the shape of a cube (m = 100 kg and side length = 42.0 cm) is released at the top of the oil layer. Where does the block come to rest? Justify your answer. If it comes to rest between two layers, specify which layers and what portion of the block sits in each layer. [Note: Vcube = a ]3 C. A small 1.00 cm radius opening is made in the side of the tank 0.500 m up from its base (block was removed). What volume of water drains from the tank in 10.0 s? (b) (a) 3. A tube is inserted into a vein in the wrist of a patient in a reclined position on a hospital bed. The heart is vertically 25.0 cm above the position of the wrist where the tube is inserted. Take DBLOOD = 1.06 × 103 kg/m3. The gauge venous blood pressure at the level of the heart is 6.16 × 103 N/m2. Assume blood behaves as an ideal nonviscous fluid. A. What is the gauge venous blood pressure at the position of the wrist? B. The tube coming from the wrist is connected to a bottle of whole blood the patient needs in a transfusion. See above figure (b). What is the minimum height above the level of the heart at which the bottle must be held to deliver the blood to the patient? C. Suppose the bottle of blood is held 1.00 m above the level of the heart. Assume the tube inserted in the wrist has a diameter of 2.80 mm. What is the velocity, v, and flow rate of blood as it enters the wrist. You may also assume the rate at which the blood level in the bottle drops is very small. The answer you get here is a substantial overstatement. Blood is not really a non-viscous fluid. 4. A 0.500 kg block is attached to a horizontal spring and oscillates back and forth on a frictionless surface with a frequency of f = 3.00 hz. The amplitude of this motion is 6.00 × 10 m. Assume to = 0 and is the instant the block is -2 at the equilibrium position moving to the left. A. Write expressions x(t) = !A sin (Tt) and v(t) = !AT cos (Tt) filling in the values of A and T. B. What is the total mechanical energy (METOT) of the block-spring system? C. Suppose the block, at the moment it reaches its maximum velocity to the left splits in half with only one of the halves remaining attached to the spring. What are the amplitude and frequency of the resulting oscillations? D. Suppose, instead of splitting at the position of maximum velocity to the left, the block now splits when it is at the extreme position in the left. What are the amplitude and frequency of the resulting motion? E. Describe in words what would happen to the period of oscillation if a second block identical to the first block were dropped on the first block at either of its extreme positions. 5. A. A spring has one end attached to a wall and the other end attached to two identical masses, mA and mB. The system is set into oscillation on a frictionless surface with amplitude A. See figure. When the system is momentarily at rest at x = -A whatever it is that holds mA to mB fails; and later in the motion mB moves away from mA to the right. 1. Location where the acceleration of mA is maximum and negative. 2. Location where the KE of mA is maximum. The next few questions ask you to compare the behavior of the mass-spring system after and before mB detached. Energy considerations are most useful here. 3. The amplitude of the mass-spring oscillation has (increased, decreased, not changed) after mB detaches. 4. The frequency of the mass-spring oscillation has (increased, decreased, stayed the same) after mB detaches. 5. The maximum speed of mA has (increased, decreased, stayed the same) after mB detaches. 6. The period of oscillation of the mass-spring system has (increased, decreased, stayed the same) after mB detaches. 7. The fraction of the total mechanical energy of the entire spring-2 mass system carried away with mB after mB detaches is B. A spherical object is completely immersed in a liquid and is neutrally buoyant some distance above the bottom of the vessel. See figure. The upper surface of the liquid is open to the earth’s atmosphere. 1. How is the density of the fluid related to the density of the spherical object? 2. Assume the fluid and object are incompressible. In addition, the $sphere (coefficient of volume expansion) > $liquid. For the following items below, indicate whether the object sinks to the bottom, rises to the surface, or does nothing based on the changes described. a. Atmospheric pressure drops by 20%. b. Salt is dissolved in the liquid in the same way fresh water is turned into salt water. c. The entire apparatus is warmed 10oC (liquid and object are both warmed). d. The entire apparatus is transported to the surface of the moon. (gmoon = 1.6 m/s , PATM = 0 on moon) 2 e. 100 cm3 of the liquid is removed from the top. The object is still initially submerged. 6. A. A mass m is attached to a spring and oscillating on a frictionless, horizontal surface. See figure. At the instant the mass passes the equilibrium position moving to the right, half the mass detaches from the other half. The oscillating system is now the spring and half the original mass with the detached mass moving off to the right with constant velocity. Relative to the original spring-mass system, the new spring-mass system with half the mass oscillates with … In the spaces provided below, enter the words larger, smaller or the same that best completes the above sentence.. 1. amplitude 2. period 3. frequency 4. maximum velocity 5. mechanical energy B. A solid cylinder is floating at the interface between water and oil with 3/4 of the cylinder in the water region and 1/4 of the cylinder in the oil region. See figure. Select the item in the parenthesis that best fits the statement. 1. The item (oil, water, and/or cylinder) with the largest density. 2. The item (oil, water, and/or cylinder) with the smallest density. 3. The weight of the cylinder (is equal to, greater than or less than) the total buoyant force it feels. 4. The density of the cylinder (is equal to, less than, or greater than) the density of water. rC. Three thermometers in different settings record temperatures T1 = 1000°F, T2 = 1000°C, and T3 = 1000 K. In the space below select T1, T2 or T3, that best fits the statement. 1. The thermometer in the hottest environment. 2. The thermometer in the coolest environment. 3. The thermometer reading a temperature 900° above the boiling point of water. 7. An oil tanker in the shape of a rectangular solid is filled with oil (Doil = 880 kg/m ). The flat bottom of the 3 hull is 48.0 m wide and sits 26.0 m below the surface of the surrounding water. Inside the hull the oil is stored to a depth of 24.0 m. The length of the tanker, assumed filled with oil along the entire length, is 280 m. View from Rear View from Side Note: Dsalt water = 1.015 × 10 kg/m ; Vrectangular solid = length × width × height. 3 3 A. At the bottom of the hull, what is the water pressure on the outside and the oil pressure on the inside of the horizontal bottom part of the hull? Assume the Po above the oil is the same as the Po above the water and its value is Po = 1.01 × 10 N/m . 5 2 B. If you did part A correctly you determined that the water pressure on the horizontal bottom part of the hull is larger than the oil pressure there. Explain why this MUST be the case. C. What buoyant force does the tanker feel? D. What is the weight of the tanker, excluding the weight of the oil in the hull? 8. A. Water is poured into a tall glass cylinder until it reaches a height of 24.0 cm above the bottom of the cylinder. Next, olive oil (Doil = 920 kg/m ) is very carefully added until the total amount of 3 fluid reaches 48.0 cm above the bottom of the cylinder. Olive oil and water do not mix. See figure. Take Dwater = 1.00 × 10 kg/m and Patm = 1.01 × 10 N/m . 3 3 5 2 1. Indicate on the drawing which layer is water and which is olive oil. 2. What is the gauge pressure 10.0 cm below the top of the upper fluid layer in the cylinder. 3. What is the gauge pressure on the bottom of the cylinder? 4. If the cylinder is in the shape of a right circular cylinder with radius of 3.60 cm, what force is exerted on the bottom of the cylinder? B. A 0.200 kg mass is hung from a massless spring. At equilibrium, the spring stretched 28.0 cm below its unstretched length. This mass is now replaced with a 0.500 kg mass. The 0.500 kg mass is lowered to the original equilibrium position of the 0.200 kg mass and suddenly released producing vertical SHM. 1. What is the spring constant for this spring? 2. What is the period of oscillation for the 0.500 kg/spring system? 3. What is the amplitude of this oscillation? r9. The drawing shows a possible design for a thermostat. It consists of an aluminum rod whose length is 5.00 cm at 20.0°C. The thermostat switches an air conditioner when the end of the rod just touches the contact. The position of the contact can be changed with an adjustment screw. What is the size of the spacing such that the air conditioner turns on at 27.0°C. This is not a very practical device. Take “al = 2.3 × 10 /°C. -5 r10. The following is an effective technique for determining the temperature TF inside a furnace. Inside the furnace is 100 gm of molten (i.e., in a liquid state) lead (Pb). The lead is dropped into an aluminum calorimeter containing 200 gm water both at an initial temperature of 10.0°C. After equilibrium is reached, the temperature reads 21.8°C. Assumptions: (1) No water is vaporized; (2) no heat is lost to or gained from the environment; and (3) the specific heat for the lead is the same whether the lead is a solid or a liquid. DATA TABLE LEAD CALORIMETER WATER mPb = 100 gm mAl = 150 gm mW = 200 gm CPb = 0.0305 cal/gm°C CAl = 0.215 cal/gm°C CW = 1.0 cal/gm°C LF = 6.0 ca./gm (heat of fusion) Tinit = 10.0°C Tinit = 10.0°C MPPb = 327°C (melting point) TF = unknown Tequilibrium = 21.8°C A. In words, describe the distinct steps in the cooling of lead. B. How many calories of heat are absorbed by the calorimeter and the water it contains to reach 21.8°C? C. How many calories are lost by the lead in cooling from TF to the final equilibrium temperature of 21.8°C? D. What was the original furnace temperature? E. If the same amount of aluminum (CAl = 0.215 cal/gm°C and LM = 21.5 cal/gm) were used in the same furnace instead of lead, would the final equilibrium temperature be higher, less or the same as in the lead case? No calculation is needed to answer this. Please explain. r11. The length of aluminum cable between consecutive support towers carrying electricity to a large metropolitan area is 180.00 m on a hot August day when the temperature is 38°C. Use “(Al) = 24 × 10-6/°C. A. What is the length of the same section of aluminum cable on a very cold winter day when T = -24°C? B. If the same length of copper (” = 17 × 10-6/°C) cable (i.e., 180.00 m on the same hot August day) were used instead of aluminum, would the length of the copper cable be shorter, longer or the same as that of the aluminum on the same winter day as in (A)? Please explain your conclusion You do not have to do any calculations here. r12. You wish to make a cup of coffee with cream in a 0.250 kg mug (cmug = 900 J/kg°C) with 0.325 kg coffee (ccoffee = 4.18 × 10 J/kg°C) starting at 25.0°C and 0.010 kg cream (ccream = 3.80 × 10 J/kg°C) at 10.0°C. 3 3 You use a 50.0 W electric heater to bring the coffee, cream and mug to a final temperature of 90.0°C. How long must the coffee system be heated? Indicate clearly the assumptions you need to make. r13. A 75.0 kg patient is running a fever of 106°F and is given an alcohol rubdown to lower his body temperature. Take the specific heat of the human body to be Cbody = 3.48 × 10 J/kg°C, the heat of 3 evaporation of the rubbing alcohol to be Lv(alcohol) = 8.51 × 10 J/kg, and the density of the rubbing 5 alcohol to be 793 kg/m3. You may assume that all the heat removed from the fevered body goes into evaporating the alcohol, and that while the patient’s body is cooling, his metabolism adds no measurable heat. A. What quantity of heat must be removed from the body to lower its temperature to 99.0°F? B. What volume of rubbing alcohol is required? C. This is a qualitative question. Give an answer and explanation. Suppose you were told that the alcohol applied started at room temperature (. 70°F) and were given the specific heat for the alcohol. Thus, you now expect some of the body heat warming the alcohol to the temperature of the fever before evaporation occurs. How would this effect the result of the calculation in part (B)? r14. A 56.0 kg hypothermia victim is running a body temperature of 91.0°F. The victim is far away from any immediate medical treatment. Her friends decide to treat the hypothermia victim by placing the victim in a sleeping bag with one of her friends and use the heat from the friend to raise the victim’s body temperature. Take the specific heat of the human body to be Cbody = 3.48 × 10 J/kg°C. Assume that the sleeping bag acts 3 like a perfect calorimeter and also assume no heat is lost to or obtained from the sleeping bag. Finally, assume all the heat that warms the hypothermia victim comes from the basic metabolic heat produced by the body of the victim’s friend in the sleeping bag with her and that metabolism is rated at 2.00 × 106 cal/day, and that the victim’s metabolism is negligible. A. How much heat must be added to the victim’s body to get her temperature up to 98.0°F? B. How long must the victim remain in the sleeping bag with her friend to achieve this temperature change? C. This is a qualitative question. If the thermal characteristics of the sleeping bag are now taken into account, but still assuming no heat leaves or enters the sleeping bag, how will the answer to question (b) above be different? r15. A few years back a lawsuit was filed by a woman against McDonald’s because she scalded herself with a Styrofoam cup filled with coffee which she spilled on herself while driving. This question was spawned by that incredible legal action and represents a possible action taken by McDonald’s to insure cooler coffee. Suppose a typical cup of coffee sold by McDonald’s is basically 400 ml of hot water and when poured into the Styrofoam cup its temperature is 96.0°C. Take 1.00 ml to have a mass of 1.00 gm and = 4.19 kJ/kg°C. Neglect any heat lost to the cup and assume no heat is lost by the coffee to the environment. A. How much heat in joules must the coffee lose to bring its temperature to a drinkable 68.0°C? B. McDonald’s possible approach to lowering the temperature of the 96.0°C coffee to 68.0°C is to add a cube of ice initially at 0.0°C. (Take Lf = 334 kJ/kg.) What mass of ice has to be added to the coffee to reduce its initial temperature to the desired 68.0°C? r16. During this past Thanksgiving your instructor overdid it and consumed 3000 Cal of food and dessert. Remember 1.0 Cal = 4.19 x 10 J. For the questions below, as 3 sume no heat is lost to the environment. [Note: = 33.5 x 105 J/kg; = 4.19 x 103 J/kgoC] A. If all of this energy went into heating 65.0 kg water starting at 37.0oC (a mass approximately that of your instructor), what would be the final temperature of this water? B. Assume your instructor removes these overeating calories by running 10 kilometer races [note: 1.61 km = 1.00 mile]. Using the rule of thumb that 1 mile of jogging will require 100 Cal, what is the minimum number of races your instructor must run to consume the 3000 Cal in part A as exercise? C. The year before, your instructor was particularly gluttonous and consumed 5000 Cal. Assuming the same conditions of water mass (65.0 kg) and starting temperature (37.0oC) as in A, what is the final temperature of the water system, and if any water vaporizes to steam, how much? [Note: BP(H2O) = 100 C] o 17. Below is the position vs. time graph for the simple harmonic of a spring oscillation on a frictionless horizontal surface. Motion to the right is positive. 1. The earliest instant of time, including t0 = 0 at which the PEelastic is maximum. 2. The earliest instant of time at which the KE of the mass is a maximum and the mass is moving to the right. 3. The earliest instant of time at which the acceleration of the mass is maximum and positive. 4. The earliest instant of time at which the speed of the mass is zero. 18. A. A spring is attached to a post at the top of a 15.0° frictionless ramp. A 2.00 kg mass is attached to the spring and the mass is slowly allowed to stretch the spring to the equilibrium position of the mass-spring system, the spring stretches by 0.400 m See figure. The mass is now pulled an additional 10.0 cm and released. The mass-spring system executes simple harmonic motion. 1. What is the spring constant, k, of the spring. 2. What are the amplitude and period of oscillation of the mass-spring system? B. A solid, uniform cylinder is floating at the interface between water (Dwater = 1.00 × 103 kg/m ) and oil (Doil = 8.24 × 10 kg/m ) with 3/4 of the cylinder in the water region and 3 3 3 1/4 of the cylinder in the oil region. Assume the axis of the cylinder is perfectly vertical. See figure. 1. What is the density of the material out of which the cylinder is made? 2. Assume the upper surface of the oil region si open to the atmosphere (Datm = 1.01 × 10 N/m ) and the oil-water interface is 0.500 m below the 5 2 upper surface of the oil. Also assume the height of the cylinder is 10.0 cm. What is the gauge pressure on the bottom surface of the cylinder? Recall: Pgauge = P – PATM. 19. A. A mass m is attached to a spring and is oscillating on a frictionless horizontal surface (see figure). At the instant the mass is at an amplitude position a second identical mass is carefully placed on top of the original mass. The oscillating system is now the spring and the two identical masses. Relative to the original spring-single mass system, the new spring-2-mass system oscillates with a … In the spaces provided below, enter (I) for increased, (D) for decreased, or (R) remains unchanged, that best completes the above last sentence. 1. amplitude. 2. period. 3. frequency. 4. spring constant. 5. maximum speed. 6. mechanical energy. 7. maximum acceleration. B. Suppose you are asked about the absolute pressure at some depth h below the surface of a liquid. The top surface is exposed to the atmosphere on a sunny day in Salt Lake City. For each statement below in the spaces provided, enter I for increase, D for decrease, or R for remains the same, when accounting for what happens to the absolute pressure at the point you are observing. 1. More liquid is added so now the observation point is farther below the surface. 2. The fluid is now exchanged for a less dense fluid. The observation point is at same h. 3. The experiment is moved to New York City, which is at sea level, on a sunny day. 4. The fluid is now seen to be moving with some speed v past the observation point. 5. The observation point is moved closer to the surface of the liquid. 6. The air above the fluid is removed by a vacuum system. 7. The apparatus is moved to a laboratory on the surface of the moon. 20. A 3.00 kg mass is attached to a spring (k = 52.0 N/m) that is hanging vertically from a fixed support. The mass is moved to a position 0.800 m lower than the unstretched position of the end of the spring. The spring is then released and the mass-spring system executes SHM. Take the 0.800 m of the mass as the reference location for its gravitational PE. A. What is the equilibrium position of the mass-spring system? B. What is the amplitude of the SHM the mass-spring system executes? C. What is the period of the oscillation of this system? D. What is the total mechanical energy of the mass-spring system at the moment the mass is released? E. What are (i) the KE of the mass and (ii) the speed of the mass when the spring is at its equilibrium position? 21. A 38.0 kg block is moving back and forth on a frictionless horizontal surface between two springs. The spring on the right has a force constant kR = 2.50 × 10 N/m. When the block is between the two 3 springs its speed (v) is 1.82 m/s. See figure. A. If the block compresses the left spring to 5.62 cm beyond its uncompressed length, determine the value of kL. B. What is the maximum compression of the right spring when the mass interacts with it? C. What is the total time the spring on the right is compressed during a single event? 22. Two identical containers are connected at the bottom via a tube of negligible volume and a valve which is closed. Both containers are filled initially to the same height of 1.00 m, one with chloroform (DC = 1530 kg/m ) in the left chamber and the other 3 with mercury in the right chamber (DHg = 1.36 × 10 kg/m ). 4 3 Sitting on top of each identical circular container is a massless plate that can slide up or down without friction and without allowing any fluid to leak past. The radius of the circular plate is 12.0 cm. The valve is now opened. A. What volume of mercury drains into the chloroform container? (Note: Vcyl = Br h) 2 B. What mass must be placed on the plate on the chloroform side to force all the mercury, but none of the chloroform, back to the mercury chamber? 23. A 12.0 kg mass M is attached to a cord that is wrapped around a wheel in the shape of a uniform disk of radius r = 12.0 cm and mass m = 10.0 kg. The block starts from rest and accelerates down the frictionless incline with constant acceleration. Assume the disk axle is frictionless. Note: Idisk = 1/2 mr . 2 A. Use energy methods to find the velocity of the block after it has moved 2.00 m down the incline. B. What is the constant acceleration of the block and the angular acceleration of the wheel? C. How many revolutions does the wheel turn for the distance the block travels in (A)? D. If the uniform disk were replaced by a uniform sphere with the same r and m of the disk, would the acceleration of the block attached to the sphere be larger, smaller, or the same as that for the block attached to the disk? Note: Isphere = 2/5 mr . 2 24. A pulley is in the shape of a uniform disk of mass m = 5.00 kg and radius r = 6.40 cm. The pulley can rotate without friction about an axis through the center of mass. A massless cord is wrapped around the pulley and connected to a 1.80 kg mass. The 1.80 kg mass is released from rest and falls 1.50 m. See figure. Note: Idisk = 1/2 mr . 2 A. Use energy methods to determine the speed of the block after falling 1.50 m. B. What is the constant acceleration of the block and the angular acceleration of the wheel? C. How many revolutions does the pulley disk turn for the distance the block travels in (A)? D Suppose the disk were replaced by a uniform sphere with the same r and m of the disk. Would the acceleration of the block attached to the sphere be larger, smaller, or the same as that for the block attached to the the disk? Note: Isphere 2/5 mr . 2 26. A 700.0 N fisherman is walking toward the edge of a 200 N plank as shown. He has placed a can of worms weighing 75.0 N on the left side of the plank as indicated in the drawing. The plank is the horizontal section in the drawing. A. Identify all the forces the plank feels before it begins to tip. Draw a free body diagram. B. As the fisherman nears the point on the plank where it begins to tip, how do the upward forces the supports exert on the plank change. C. How far a distance, as measured from the center of the right support, can he walk before the plank begins to tip? 26. A 75.0 kg sign hangs from a 4.80 m uniform horizontal rod whose mass is 120 kg. The rod is supported by a cable that makes an angle of 53° with the rod. he sign hangs 3.60 m out along the rod. A. What is the tension in the cable? B. What are the forces PPv and PPH exerted by the wall on the left end of the rod? 27. A 1.00 × 104 N great white shark is hanging by a cable attached to a 4.00 m massless rod that can pivot at its base. See figure. A. Determine the tension in the cable supporting the upper end of the rod. See figure. B. Determine the force (a vector quantity) exerted on the base of the rod. Suggestion: Find this force by first evaluating the separate components of the force. See figure. 28. A 6.00 m uniform beam extends horizontally from a hinge fixed on a wall on the left. A cable is attached to the right end of the beam. The cable makes an angle of 30.0° with respect to the horizontal and the right end of the cable is fixed to a wall on the right. At the right end of the cable hangs a 140.0 kg mass. The mass of the beam is 240.0 kg. See figure. A. Find the tension in the cable. B. Find the vertical and horizontal forces the hinge exerts on the left end of the beam. 29 A. The blades of a “Cuisinart” blender when run at the “mix” level, start from rest and reach 2.00 × 103 rpm (revolutions per minute) in 1.60 s. The edges of the blades are 3.10 cm from the center of the circle about which they rotate. 1. What is the angular acceleration of the blades in rad/s2 while they are accelerating? 2. Through how many rotations did the blades travel in that 1.60 s? 3. If the blades have a moment of inertia of 5.00 × 10-5 kg m2, what net torque did the blades feel while accelerating? B. A 7.50 × 10 N 4 shipping crate is hanging by a cable attached to a uniform 1.20 × 104 N steel beam that can pivot at its base. A second cable supports the beam and is attached to a wall. See figure. 1. Determine the tension T in the upper cable. 2. Determine the magnitude of the force exerted on the beam at its base. See drawing. 30. The drawing shows a uniform ladder of length L and weight 220 N. The ladder is sitting at an angle of 30° above the horizontal resting on the corner of a concrete wall at a point that is one-fourth of the way from the end of the ladder. A 640 N construction worker is standing on the ladder one-third of the way up from the end of the ladder which is resting on the ground. Assume the corner of the wall on which the ladder rests exerts only a normal force on the ladder at the point where there is contact. A. What is the magnitude of the normal force the wall exerts on the ladder? B. Find the magnitude of both the normal force the ground exerts on the left end of the ladder and the static frictional force the ground exerts on the left end of the ladder. 31. A. A solid, right circular cylinder (radius = 0.150 m, height = 0.120 m) has a mass m. The cylinder is floating in a tank in the interface between two liquids that do not mix: water on the bottom and oil above. One-third of the cylinder is in the oil layer (Doil = 725 kg/m ) 3 and two-thirds in the water layer (Dwater = 1.00 × 10 kg/m ). See 3 3 drawing. Note: V(circular cylinder) = B r2 h. 1. Find the mass of the cylinder. 2. With the cylinder present, take the thickness of the oil layer to be 0.200 m and the thickness of the water layer to be 0.300 m. What is the gauge pressure at the bottom of the tank? Assume the top of the oil layer is exposed to the atmosphere. B. A block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of T = 7.52 rad/s. The drawing indicates the position of the block when the spring is unstretched. That position is labeled “x = 0 m” in the drawing. The drawing also shows a small bottle whose left edge is located at Xb = 0.0900 m. The block is now pulled to the right, stretching the spring by Xs = 0.0343 m, and is then thrown to the left, i.e., given an initial push to the left. In order for the block to knock over the bottle when it is moving to the right, it must be “thrown” with an initial speed to the left v0. Ignoring the width of the block, what is the minimum value of v0? 32. B. Three objects, a disk (ICM = ½ MR ), a hoop (ICM = MR ), and a hollow ball (ICM = b MR ) all have 2 2 2 the same mass and radius. Each is subject to the same uniform tangential force that causes the object, starting from rest, to rotate with increasing angular speed about an axis through the center of mass for each object. In the case of the hollow ball the tangential force has a moment arm equal to the radius of the ball. In the space below, enter D for disk, H for hoop, and/or B for hollow ball, or same to best answer the question. 1. The object with the largest moment of inertia about the axis through the CM. 2. The object experiencing the greatest net torque. 3. The object with the greatest angular acceleration during the period the force is acting. 4. The object rotating with the smallest angular speed assuming the force has been acting for the same length of time on each object. 33. A. A uniform disk (D), hoop (H), and sphere (S), all with the same mass and radius, can freely rotate about an axis through the center of mass (CM) of each. A massless string is wrapped around each item. The string is used to apply a constant and equal tangential force to each object. See figure. For the statements below, enter D, H, S, none or the same. Assume all objects start from rest at the same instant. 1. The one with the smallest moment of inertia about the shown axis. 2. The object experiencing the largest net torque. 3. The object undergoing the smallest angular acceleration. 4. The object with the largest angular speed after an elapsed time of 5.0 s. 5. The object for which the largest amount of string has unraveled in 5.0 s. 6. The object with the smallest KErot after 5.0 s. 7. The object that undergoes the most rotations in 5.0 s. B. A spherical object is completely immersed in a liquid of density Dliq some distance above the bottom of the vessel. See figure. The upper surface is initially open to the earth’s atmosphere at sea level. Assume the liquid and object are both incompressible. For the items below, indicate whether the object sinks to the bottom (B), rises to the surface (T), or does nothing (N). 1. The vessel is brought to Salt Lake City. 2. Salt is dissolved in the liquid in the same way fresh water is turned into salt water. 3. The top 50 cm3 of the liquid is removed from the vessel. 4. The entire apparatus is transported to the surface of the moon. 5. The volume of the spherical object is increased by heating it without heating the liquid. 6. The spherical object is moved 10 cm farther down in the vessel and released. 7. A mass is placed on the top surface of the liquid in the vessel increasing the pressure at the surface. No fluid leaks. 34. A 2.20 × 103 N uniform beam is attached to an overhead beam as shown in the drawing. A 3.60 × 103 N trunk hangs from an attachment to the beam two-thirds of the way down from the upper connection of the beam to the overhead support. A cable is tied to the lower end of the beam and is also attached to the wall on the right. A. What is the tension in the cable connecting the lower end of the beam to the wall? B. What are magnitude of the vertical and horizontal components of the force the overhead beam exerts on the upper end of the beam at P? 35. A. A 12.0 kg block moves back and forth on a frictionless horizontal surface between two springs. The spring on the right has a force constant k = 825 N/m. When the block arrives at the spring on the right, it compresses that spring 0.180 m from its unstretched position. 1. What is the total mechanical energy of the block and two spring system? 2. With what speed does the block travel between the two springs while not in contact with either spring? 3. Suppose the block, after arriving at the left spring, remains in contact with that spring for a total time of 0.650 s, before separating on its way to the right spring? Using the connection between this 0.650 s and the period of oscillation between the block and the left spring, determine the spring constant of the left spring. B. A turkey baster (see figure) consists of a squeeze bulb attached to a plastic tube. When the bulb is squeezed and released, with the open end of the tube under the surface of the turkey gravy, the gravy rises in the tube to a distance h, as shown in the drawing. It can then be squirted over the turkey. Using Patm = 1.013 × 105 N/m2 for atmospheric pressure and 1.10 × 103 kg/m3 for the density of the gravy, determine the absolute pressure of the air in the bulb with the distance h = 0.160 m. Give answer to three significant digits. 36. A. The pictures below depict three glass vessels, each filled with a liquid. The liquids each have different densities, and DA > DB > DC. In vessel C an unknown block is neutrally buoyant halfway to the bottom and completely submerged. A, B, and/or C, or none are all possible answers. 1. _______ In which vessel(s) would the block sink all the way to the bottom? 2. _______ In which vessel(s) would the largest volume of the block be exposed above the surface of the liquid? 3. _______ In which vessel(s) would the buoyant forces on the block be the same? B. A swinging pendulum (A) and a mass-spring system (B) are built to have identical periods. For the statements below enter either A, B, U (unchanged) to best fit which oscillating system would have the larger period as a result of the change. 1. _______ The mass of the mass-spring system is increased. 2. _______ The mass of the swinging pendulum is increased without altering the location of its center of mass. 3. _______ The spring constant of the mass-spring system is increased. 4. _______ The length of the swinging pendulum system is increased. 5. _______ Both systems are taken to the moon and set oscillating. C. A block of mass m moves back and forth on a frictionless surface between two springs. See drawing. Assume kL > kR. For the statements below enter L for the left spring, R for the right spring, or same as the case may be. 1. _______ The spring that has the maximum compression when m is momentarily at rest. 2. _______ The spring that stores the larger elastic potential energy when maximally compressed. 3. _______ The spring that momentarily stops the block in the least time once the block arrives at the spring. 37. A uniform beam extending at right angles from a wall is used to display an advertising sign for an eatery. The beam is 2.50 m long an weighs 80.0 N. The sign, whose dimensions are 1.00 m by 0.800 m, is uniform, and weighs 200. N, hangs from the beam as shown in the drawing. A cable, attached to the wall of the eatery at a point on the beam where the inside end of the sign is attached to the beam and making an angle of 60.0° with the beam, supports this advertising structure. A. What is the magnitude of the tension in the cable supporting the beam? B. What are the magnitudes of the horizontal and vertical forces the wall exerts on the left end of the beam? 38. A. Examine the picture shown to the right. Initially, before the pump is turned on, the two masses (m1 = 1.00 kg, m2 = 2.75 kg) are held in place. the pressures above and below m1 are Patm = 1.01 × 10 N/m and 5 2 the spring is in its unstretched position. The pump is turned on and the masses are allowed to move. The mass m1 moves without friction inside a cylindrical piston of radius r = 3.85 cm. Once equilibrium is established, by what distance has the spring stretched? Take k = 2.00 × 103 N/m for the spring constant. B. A solid cylinder (radius 0.125 m and height 0.150 m) has a mass of 6.50 kg. The cylinder is floating in water. Oil (Doil = 725 kg/m ) is poured on top of the water until 3 the situation shown in the drawing results. How much of the height (in meters) of the cylinder remains in the water layer?

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## Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

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