# Potential Energy & Conservation of Energy

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1 PHYS 101 Previous Exam Problems CHAPTER 8 Potential Energy & Conservation of Energy Potential energy Conservation of energy conservative forces Conservation of energy friction Conservation of energy external forces 1. A 2.0-kg block is released from rest 60 m above the ground. Take the gravitational potential energy of the block to be zero at the ground. At what height above the ground is the kinetic energy of the block equal to half its gravitational potential energy? (Ignore air resistance) (Ans: 40 m) 2. A projectile is fired from the top of a 40-m high building with a speed of 20 m/s. What will be its speed when it strikes the ground? (Ans: 34 m/s) 3. A 6.0-kg box starts up a 30 degrees incline with 158 J of kinetic energy. How far will it slide up the incline if the coefficient of kinetic friction between box and incline is 0.40? (Ans: 3.2 m) 4. Figure 1 shows a pendulum of length L = 1.0 m. Its ball has speed v o = 2.0 m/s when the cord makes an angle of 30 o with the vertical. What is the speed of the ball when it passes the lowest position? (Ans: 2.6 m/s) 5. A 0.6-kg ball is suspended from the ceiling at the end of a 2.0-m string. As this ball swings, it has a speed of 4.0 m/s at the lowest point of its path. What maximum angle does the string make with the vertical as the ball swings? (Ans: 54 o ) 6. An object of mass m, attached to a light cord of length L, is held horizontally from a fixed support as shown in figure 2. The object is then released from rest. What is the tension force in the cord when the object is at the lowest point of its swing? (Ans: 3 mg) 7. A ball slides without friction around a loop-the-loop of radius R (see figure 3). The ball is released, from rest, at a height h from the left side of the loop. What is the ratio (h/r) so that the ball has a speed v = Rg at the highest point of the loop? (Ans: 5/2) 8. An ideal spring with a 20-N/m spring constant is compressed by a 10 N force. What is the potential energy stored in the spring? (Ans: 2.5 J) 9. A 0.50-kg block attached to an ideal spring with a spring constant of 80 N/m oscillates on a horizontal frictionless surface. The speed of the block is 0.50 m/s when the spring is stretched by 4.0 cm. What is the maximum speed the block can have? (Ans: 0.71 m/s) 10. A 4.0-kg block is initially moving to the right on a horizontal frictionless surface at a speed of 5.0 m/s. It then compresses a horizontal spring (k = 200 N/m). At the instant when the kinetic energy of the block is equal to the potential energy of the spring, what is the mechanical energy of the block-spring system? (Ans: 50 J) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 1

2 11. A 12-kg block is resting on a horizontal frictionless surface. The block is attached to an un stretched spring (k = 800 N/m) (see figure 4). A force F = 80 N parallel to the surface is applied to the block. What is the speed of the block when it is displaced by 13 cm from its initial position? (Ans: 0.78 m/s) 12. A 2.0-kg block slides on a rough horizontal table top. Just before it hits a horizontal ideal spring its speed is 5.0 m/s. It hits the spring and compresses it 10.0 cm before coming momentarily to rest. If the spring constant is 1200 N/m, how much work is done by friction? (Ans: 19 J) 13. A 3.0-kg block is released from a compressed spring (k = 120 N/m). It travels over a horizontal surface (μ k = 0.20) for a distance of 2.0 m before coming to rest (see figure 5). How far was the spring compressed before being released? (Ans: 0.44 m) 14. A block (mass = kg) is pushed against a vertical spring compressing the spring a distance of h = 8.00 cm (see figure 6). The block is not attached to the spring. When released from rest, the block rises to a maximum height of H = m. Calculate the spring constant. (Ans: 184 N/m) 15. A 2.0-kg object is connected to one end of an unstretched spring, which is attached to the ceiling by the other end; and then the object is allowed to drop. The force constant of the spring is 196 N/m. How far does it drop before coming momentarily to rest? (Ans: 0.20 m) 16. A 2.2-kg block starts from rest on a rough inclined plane that makes an angle of 30 above the horizontal. The coefficient of kinetic friction is As the block moves 3.0 m down the plane, what is the change in the mechanical energy of the block? (Ans: 14 J) 17. A 5.0-kg block starts up a 30 incline with 198 J of kinetic energy. The block slides up the incline and stops after traveling 4.0 m. What is the work done by the force of friction? (Ans: 100 J) 18. A 75-kg parachutist releases himself off a tower that is 85 m high. Assume that he starts from rest and reaches the ground with a speed of 5.0 m/s. How much work was done by the non conservative forces on him? (Ans: J) 19. As a particle moves from point A to point B only two forces act on it: one force is non-conservative and does work = 30 J, the other force is conservative and does + 50 J work. What is the change in the kinetic energy of the particle? (Ans: 20 J) 20. A 0.50-kg block attached to a spring, with a spring constant of 100 N/m, moves on a horizontal surface having a coefficient of kinetic friction of 0.3 (see figure 7). The spring is initially compressed by 10 cm from the unstretched position O and then released from rest. What is the speed of the block when it passes through the point O? (Ans: 1.2 m/s) 21. A ball of mass 2.0-kg is kicked with an initial velocity of v = 5 î + 5 ĵ (m/s). What is the ratio of the potential energy (relative to ground level) to the kinetic energy of the projectile at its highest point? (Ans: 1.0) 22. A 4.0-kg cart starts up an incline with a speed of 3.0 m/s and comes to rest 2.0 m up the incline. What is the net work done on the cart? (Ans: -18 J) 23. A block, of mass m = 200 g, slides back and forth on a frictionless surface between two springs, as shown in figure 18. The left-hand side spring has k 1 = 130 N/m and its maximum compression is 16 cm. The right-hand side spring has k 2 = 280 N/m. Find the maximum compression of the right-hand side spring. (Ans: 11 cm) 24. A block of mass 2.0 kg is initially moving to the right on a horizontal frictionless surface at a speed of 5.0 m/s. It then compresses a spring of spring constant 100 N/m. At the instant when the kinetic energy of the block is equal to the potential energy of the spring, by what distance is the spring compressed? (Ans: 0.50 m) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 2

3 25. A 2-kg object is dropped vertically from rest. After falling a distance of 50 m, it has a speed of 25 m/s. Calculate the work done by air resistance on the object during the fall. (Ans: -355 J) 26. A block of mass 1.6 kg, resting on a horizontal frictionless surface, is attached to a horizontal spring fixed at one end. The spring, having a spring constant of N/m, is compressed to x = -2.0 cm (x = 0.0 is the equilibrium position) and the block is released from rest. What is the speed of the block as it passes through the position x = cm? (Ans: 0.43 m/s) 27. A 0.50-kg ball is dropped vertically from rest from a height of 3.0 m and bounces back to a maximum height of 2.0 m. What is the magnitude of the energy dissipated in the collision with the floor? (Ans: 4.9 J) 28. A projectile of mass m = kg is fired at an angle of 60.0 o above the horizontal with a speed of 20.0 m/s. Find the work done on the projectile by the gravitational force during its flight from its firing point to the highest point on its trajectory. (Ans: J) 29. Figure 9 shows an object of mass m = 1.00 kg starting from rest. It first slides a distance of 45.0 cm down a frictionless inclined surface, and then slides across a rough horizontal surface whose coefficient of kinetic friction is What is the maximum distance d travelled by the object across the horizontal surface? (Ans: 103 cm) 30. A block of mass 2.00 kg is released from rest and slides down a rough track of radius R = 1.00 m, as shown in figure 11. If the speed of the block at the bottom of the track is 4.00 m/s, what is the work done by the frictional force acting on the block? (Ans: 3.60 J) 31. In figure 12, a 2.0-kg object slides on a frictionless horizontal surface toward a spring. The speed of the object just before it hits the spring is 6.0 m/s and its speed when the spring is compressed 15 cm is 3.7 m/s. Find the spring constant of the spring. (Ans: N/m) 32. A spring, with spring constant k = 106 N/m, is attached to the top of a frictionless 30 o incline, as shown in figure 13. The distance between the lower end of the incline and the relaxed end of the spring is 1 m. A 1-kg block is pushed against the spring until the spring is compressed by 0.2 m, and released from rest. Find the speed of the block when it reaches the lower end of the incline (ignore the size of the block). (Ans: 4.0 m/s) 33. A particle moves under the influence of a single conservative force. At point (A), the potential energy associated with the conservative force is +40 J. As the particle moves from (A) to (B), the force does +25 J of work on the particle. What is the value of the potential energy at point B? (Ans: +15 J) 34. The two masses in figure 14 are released from rest. After the 3.0-kg mass had fallen 1.5 m, it reached a speed of 3.8 m/s. How much work is done during this time interval by the frictional force on the 2.0- kg mass? [Assume that the pulley is frictionless and massless] (Ans: 8.0 J) 35. Two blocks A and B (M A = 50.0 kg and M B = 100 kg) are connected by a string as shown in figure 15. The pulley is frictionless and of negligible mass. Determine the change in the kinetic energy of the two-block system as block B moves down a vertical distance of 20.0 m. Assume a smooth incline. (Ans: 13.7 kj) 36. In figure 12, a block (m = 2.0 kg) slides on a frictionless horizontal surface towards a spring with a spring constant k = 2000 N/m. The speed of the block just before it hits the spring is 6.0 m/s. How fast is the block moving at the instant the spring has been compressed 15 cm? (Ans: 3.7 m/s) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 3

4 37. Figure 16 shows a 3.00 kg block moving on a rough horizontal surface. The block strikes a spring with a spring constant k = 20.0 N/m and compresses it 1.50 m before coming to rest. The coefficient of kinetic friction between the block and the surface is Calculate the speed of the block just before it strikes the spring. (Ans: 4.57 m/s) 38. As shown in figure 17, a block of mass m = 1.35 kg is held against a compressed spring of spring constant k = 560 N/m. The spring is compressed by x = m. The block is released and slides a distance d = m to point A. Find the speed of the block at point A if the coefficient of kinetic friction between the block and the surface is μ k = (Ans: 1.57 m/s) 39. A block is released from rest at the top of an inclined plane making an angle of 30.0 with the horizontal. The coefficient of kinetic friction between the block and the inclined plane is What is the speed of the block after it has traveled a distance of 1.00 m downwards along the inclined plane? (Ans: 2.17 m/s) 40. A block slides back and forth in a hemispherical bowl, starting from rest at the top point A, as shown in figure 19. The bowl is frictionless except for a 2.0 cm-wide rough flat surface at the bottom, where coefficient of kinetic friction is μ k = How many times does the block cross the rough region before coming to rest? (Ans: 10 times) 41. In figure 21, a 5.0-kg block is moving at 5.0 m/s along a horizontal frictionless surface toward an ideal spring that is attached to a wall. After the block collides with the spring, the spring is compressed a maximum distance of x m. What is the speed of the block when the spring is compressed to only x m/2? (Ans: 4.3 m/s) 42. The only force acting on a particle is a conservative force F. If the particle is at a point A, the potential energy of the system is 80 J. If the particle moves from point A to point B, the work done on the particle by F is +20 J. As the particle reaches point B, what is the potential energy of the system? (Ans: 60 J) 43. A 2.00-kg mass is moved along a rough vertical circular track (radius R = m) as shown in figure 22. The speed of the mass at point A is v A = 8.00 m/s, and at point B is v B = 5.00 m/s. How much work is done on the mass between A and B by the force of friction? (Ans: 7.64 J) 44. A 100-kg parachute descends at a constant speed of 1.00 m/s. At what rate is energy being lost? (Ans: 980 W) 45. A block of mass m = 12.0 kg is released from rest on a frictionless incline of angle θ = 30.0 (see figure 23). Below the block is a spring that can be compressed 2.50 cm by a force of 250 N. The block stops momentarily when it compresses the spring by 4.00 cm. How far does the block move down the incline from its rest position to this stopping point? (Ans: 13.6 cm) 46. A block of mass m = 10 kg is connected to an un-stretched spring (k = 400 N/m) (see figure 25). The block is released from rest. If the pulley is massless and frictionless, what is the maximum extension of the spring? (Ans: 49 cm) 47. As a particle moves along the x-axis it is acted on by a conservative force F(x). The potential energy U(x) of the particle as a function of x is shown in figure 26. What is the force F(x)? (Ans: + 10 N) 48. A massless spring has a spring constant of 500 N/m. A 2.0 kg object is released from rest at a height h = 1.0 m above the spring, and lands on it (figure 27). Find the object s speed when the spring is compressed 20 cm. (Ans: 3.7 m/s) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 4

5 49. A 15.0 kg stone slides down a smooth snow-covered hill (figure 28), leaving point A with a speed of 4.0 m/s. Then it slides a distance of 118 m on a rough horizontal surface from point B to point C before coming to rest. Find the coefficient of kinetic friction μ k between the stone and the surface. (Ans: 0.600) 50. A mass m = 1.00 kg is released from rest at point P (h = 6.00 m). It slides along the smooth track and reaches point O on the circular part of the track (R = 1.00 m) shown in figure 29. What force does the track exert on the mass at point O? (Ans: 68.6 N) 51. A block of mass 4.00 kg is moving across a rough horizontal floor where the coefficient of kinetic friction is If the block slides for 3.00 m across the floor, and the thermal energy of the block increases by 30.0 J. Find the increase in the thermal energy of the floor. (Ans: 40.6 J) 52. A 2.00 kg ball is thrown with an initial velocity of vo = 18 î + 10 ĵ (m/s). What is the maximum change in the potential energy of the ball-earth system during its flight? (Ans: 100 J) 53. A 2.00 kg package is released on a rough 53.1 o incline at 4.00 m from a long spring of force constant 120 N/m. The spring is attached to the bottom of the incline as shown in figure 31. If the maximum compression of the spring is d = 1.00 m, what is the work done by the friction force? (Ans: 18.4 J) A B C D E Conceptual Problems 1. A small object of mass m on the end of a massless rod of length L is held vertically, initially. The rod is pivoted at the other end O. The object is then released from rest and allowed to swing down in a circular path as shown in figure 8. What is the speed (v) of the object at the lowest point of its swing? (Assume no friction at the pivot) 2. As an object moves from point A to point B, only two forces act on it: one force is conservative and does 10 J of work, the other is non-conservative and does -20 J of work. What happens to the energy of the object between points A and B? A. Kinetic energy decreases, mechanical energy decreases. B. Kinetic energy decreases, mechanical energy increases. C. Kinetic energy increases, mechanical energy decreases. D. Kinetic energy increases, mechanical energy increases. E. Mechanical energy is conserved. Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 5

6 3. A ball of mass m, attached at one end of a massless string of length L, rotates in a vertical circle fast enough to prevent the string from going loose at the top of the circle. Neglecting air resistance, during this motion, which ONE of the following statements is WRONG: A. Change in mechanical energy is not zero. B. The speed of the ball at the bottom of the circle is maximum. C. The work done by the tension is zero. D. Kinetic energy is not conserved. E. Gravitational potential energy is not conserved. 4. Figure 10 shows three identical blocks that are moving on three identical rough surfaces having the same coefficient of kinetic friction. They all move the same distance parallel to the surface. Rank the 3 situations according to the magnitude of the work done by the force of friction, greatest first. A. a then b and c tie B. a then b then c C. b then a, then c D. c then b, then a E. not enough information to decide 5. Two bodies of different masses M and m are raised to the same height above the floor, and released from rest simultaneously. Assuming that air resistance is constant and identical on the two bodies, which one of the following statements is true? A. The larger mass reaches the floor first. B. The two masses reach the floor at the same time. C. The smaller mass reaches the floor first. D. There is not enough information to say which one of the two masses will reach the floor first. E. The two masses have the same speed as they reach the floor. 6. A block is moving along a frictionless horizontal track when it enters the circular vertical loop as shown in figure 20. The block passes points 1, 2, 3, 4, 1 before returning to the horizontal track. Which one of the following statements describes the block at point 3 correctly? A. Its speed is a minimum B. The forces on it are balanced C. It is not accelerating D. Its mechanical energy is a minimum E. It experiences a net upward force 7. The work done by a conservative force acting on a body A. does not change the total energy. B. does not change the potential energy. C. does not change the kinetic energy. D. is always equal to zero. E. is always equal to the sum of the changes in potential and kinetic energies. 8. In figure 24, a moving block can take three frictionless paths, differing only in elevation (height), to reach the dashed finish line. Rank the paths according to the speed of the block at the finish line, greatest first. A) 3,2,1 B) 1,2,3 C) 2,3,1 D) 1,3,2 E) 2,1,3 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 6

7 9. A light object and a heavy object are initially sliding with equal speeds along a horizontal frictionless surface. Then, they both slide up the same frictionless incline. Which object rises to a greater height? A) They both slide to the same height. B) The heavy object, because it has greater kinetic energy C) The light object, because it has smaller kinetic energy. D) The light object, because it weighs less. E) The heavy object, because it weighs more. 10. A block of mass m sliding down a rough incline (angle θ, coefficient of kinetic friction μ) at constant speed is initially at a height h. What is the increase in the thermal energy of the block-incline system when the block reaches the bottom? A. mgh B. mgh/μ C. μmgh/sinθ D. mgh cosθ E A block initially at rest is allowed to slide down a frictionless ramp of height h and attains a speed v at the bottom. To achieve a speed 2v at the bottom, how high must the new ramp be? A. 4h B. H C. 2h D. 3h E. 5h 12. As shown in figure 30 (not to the scale), a small block is released from rest on a frictionless ramp at a height of 3.0 m. The hill heights along the ramp are as shown. The hills have identical circular tops and the block does not fly off any hill. At which two hill tops values of the normal force on the block will be maximum and minimum, respectively? A. 3, 1 B. 4, 1 C. 2, 3 D. 1,4 E. 3, 2 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 7

8 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 8

9 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 9

10 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 8 Page 10

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