2. Which of the following best describes the relationship between force and potential energy?

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1 Work/Energy with Calculus 1. An object oves according to the function x = t 5/ where x is the distance traveled and t is the tie. Its kinetic energy is proportional to (A) t (B) t 5/ (C) t 3 (D) t 3/ (E) t (kinetic energy is constant). Which of the following best describes the relationship between force and potential energy? (A) Force is the anti-derivative of potential energy (C) Potential energy is the negative gradient of force (E) Force is the anti-derivative of potential energy (B) Force is the negative gradient of potential energy (D) Potential energy is the derivative of force 3. A student throws a ball of ass upward where the initial potential energy is 0. At a height H the ball has a potential energy of U and is oving upward with a kinetic energy of Ke. Ignoring air resistance, the axiu height achieved by the ball is ost nearly (A) (Ke-U)/g (B) (U-Ke)/g (C) (U+Ke)/g (D) g/(u-ke) (E) g/(u+ke) 4. A box of ass is lifted a vertical distance h in tie t with a constant velocity. The power supplied by the lifting force is approxiately (A) ght (B) gh/t (C) zero (D) gt/h (E) g/ht 5. A 5-kilogra ball oves in the x direction where x represents the ball s position. The potential energy U of the ball in joules is given as a function by: U(x) = 4x - 3x +. The force on the particle at x = 4 eters is (A) 9 N in +x direction (B) 9N in -x direction (C) 108 N in -x direction (D) 45 N in -x direction (E) 108 N in +x direction 6. A student pushes a box across a rough, flat surface at a constant speed v. The box has a ass, and the coefficient of sliding friction is represented by µ. The power supplied by the person to the box is (A) 0 (B) µg/v (C) µv/g (D) g/µv (E) µgv 7. The force exerted by a spring is given by: F=kx 4 /. If k is 100 N/ find the work done fro x=0 to x=. (A) 100 J (B) 30 J (C) 800 J (D) 1600 J (E) 400 J 8. A an lifts a ass at constant speed to a height h in tie t. How uch work is done by the weight lifter? (A) gt (B) zero (C) gh (D) gh/t (E) cannot be deterined 9. It takes a force F given by F = 0x - 1x to stretch an irregular spring by a distance x, where F is in newtons and x is in eters. What is the change in potential energy when the spring is stretched 3 eters fro its equilibriu position? (A) 18J (B) 8J (C) 56J (D) 64J (E) 80J

2 10. On top of a skyscraper of height H, a ball of ass is thrown directly downward with an initial speed vo. If the ball reaches the ground with a speed vf, the energy lost to friction is approxiately (A) gh- 1 (vo -vf ) (B) gh+ 1 (vo +vf ) (C) gh+ 1 (vf -vo ) (D) gh+ 1 (vf +vo ) (E) 1 (vf -vo )-gh 11. A ball attached to a string rotates in a coplete circle. The work done during each revolution is (A) 0 (B) U (C) U+Ke (D) Ke (E) Ke-U 1. The potential energy of two olecules is given by: U=/r 7-4/r 5. If r is the distance between two olecules what is the force acting on the particles if r=1? (A) 0.75 N (B) 0.67 N (C) N (D) 6 N (E) 10 N 13. A force of 40 N copresses a spring with a spring constant 80 N/. How uch energy is stored in the spring? (A) 10 J (B) 15 J (C) 0 J (D) 5 J (E) 30 J M1 M 14. When the object of ass 1 descends to a distance h, the potential energy of the syste has decreases by (A) (1 )gh (B) 1gh (C) (1 + )gh (D) ½(l + )gh (E) What is the power delivered by gravity to a 6kg block 4 seconds after it has fallen fro rest? (A) 400 W (B) 1000 W (C) 800 W (D) 100 W (E) 000 W 16. If F(x)=x 3-3x what is the work done fro x=1 to x=? (A) 0.5 J (B) 0.8 J (C) J (D) 1 J (E) 0 J 17. A 5 kilogra block is pushed horizontally across a rough surface with a coefficient of kinetic friction of 0. by a force F oving at 3 /s. The work that is done by the force in 0 seconds is (A) 00 J (B) 400 J (C) 600 J (D) 800 J (E) 1000 J Questions F x -4

3 18. The graph represents the force exerted on a particle. What is the work done on the object fro x=0 to x= 4? (A) 8 J (B) 10 J (C) 15 J (D) 16 J (E) 0 J 19. What is the work done on the object fro x=0 to x=6? (A) 4 J (B) 6 J (C) 10 J (D) 1.5 J (E) 5 J 0. A constant force supplies an average power of 8 watts to a box during a certain tie interval. If the box has an average speed of 4 eters per second and the force acts in the sae direction as otion of the object, the agnitude of the force is (A) 8 N (B) 6 N (C) 4 N (D) N (E) 1N Questions 1-3 A ball of ass is tossed fro roof of height h with an initial velocity vo. 1. What is the kinetic energy of the ball at the top of the roof? (A) 0 (B) gh (C) ½ vo (D) gh-½ vo (E) gh+½ vo. What is the potential energy of the ball when it hits the ground? (A) 0 (B) gh/ (C) ½ vi (D) gh/+½ vi (E) gh+½ vi 3. What is the kinetic energy of the ball right before it strikes the ground? (A) 0 (B) gh (C) ½ vi (D) gh-½ vi (E) gh+½ vi 4. The following graphs represent the net force F as a function of displaceent x for an ball that oves in a straight line. Which graph best represents the force that will cause the least change in the kinetic energy of the object fro x = 0 to x = x1? Questions 5-7 A B C A block is attached to a spring that oscillates in siple haronic otion.

4 5. At which point is the kinetic energy of the ball at axiu? (A) Point A (B) Point B (C) Point C (D) Points A&B (E) Kinetic energy is the sae at all points 6. What happens to the potential energy as the ball passes fro point A to point B? (A) increases (B) decreases (C) stays the sae (D) zero (E) not enough inforation 7. Which of the following is true about the syste? (A) Potential energy is never equal to Kinetic energy (B) Potential energy is equal to Kinetic energy at all points (C) The syste will eventually stop oving (D) There is a restoring force (E) Both B&D 8. A 4 kg ass is oving with a velocity given by v (t) = ¼t 4. At t= 1 s, the instantaneous power delivered by the net force is (A) 1 W (B) 3 W (C) 1 W (D) 14 W (E) 0 W 9. What is the instantaneous power delivered by the net force at the t = s to a kg ass oving according to x(t) = t 3 /3? (A) W (B) 10 W (C) 16 W (D) 4 W (E) 3 W Questions U(R) 4Uo Uo Ro 4Ro R A particle of ass follows the potential energy graph as shown above. The ass is initially at rest at RO. 30. Deterine the particle s speed at position 4ro (A) U 0 (B) U 0 (C) 4U 0 (D) 6U 0 (E) 8U The function U(r) = ar -3/ + b represents the potential energy of a particle, where b and c are constants, which of the following is an expression for the force on the particle? (A) 3 5/ ar (B) 3 1/ ar (C) 3 1/ ar (D) ar 1/ br (E) ar 5/ br 5 Questions 3-33

5 The force exerted by a non-unifor spring is F = -kx 3. The axiu displaceent of the spring is x =. The spring constant of the spring is 36 N/ and the ass of the block is.5 kg. 3. Find the total energy of the syste. (A) 1 J (A) 144 J (C) 36 J (D) 10 J (E) 43 J 33. Find the axiu speed of the syste. (A) 9.8 /s (B) 4 /s (C) 15 /s (D) 0 /s (E) 7 /s Question H 3 4 A block of ass slides down an incline and experiences a friction force with a coefficient of kinetic friction µ on the rap and the track it follows. The block coes to rest at point At which point is the kinetic energy the greatest? (A) Point 1 (B) Point (C) Point 3 (D) Points 4 (E) Between Point 1& 35. Which of the following best represents the energy lost to friction fro point 1 to point 4? (A) µgh (B) MgH (C) gh (D) g/h (E) gh 36. A ass oves under the influence of a potential energy given by: U(x) = x 3 x At x=, the force on the ass will be (A) 4 N, + direction (B) 4 N, - direction (C)1.3 N, + direction (D) 1.3 N, - direction (E) 7 N, + direction 37. A ass is attached to a spring on a frictionless horizontal surface. It is displaced by an aount of x eters and then released. Which of the following stateents are true? I. PE and KE will never be equal. II. KE is largest when the ass is passing through the equilibriu point. III. PE is largest when the ass has a displaceent of ±x. (A) I only (B) II only (C) III only (D) I and II only (E) II and III only 38. A 6 kg object s potential energy is represented by: U=9r +4. Find the acceleration of the object at r= (A) 1 /s (B) 4 /s (C) 6 /s (D) 9 /s (E) 1 /s

6 39. A restoring force F = -x 3 acts on an object, where x is the displaceent of the object fro its equilibriu position. How uch work ust be done to ove the object fro x= 0 to x=? A) J B) 4 J C) 8 J D) 10 J E) 15 J Questions F(x) = sin x F 40. Find the work done fro x = 0 to x = π/. (A) 0 J (B) 1 J (C) π/ J (D) 5 J (E) 10 J 41. Find the work done fro x = 0 to x = π. (A) 0 J (B) 1 J (C) 1.5 J (D) -1 J (E) 4 J Answers 1c b 3c 4b 5b 6e 7b 8c 9a 10c 11a 1d 13a 14a 15d 16a 17c 18a 19b 0d 1c a 3e 4c 5b 6b 7d 8a 9e 30d 31a 3b 33b 34b 35b 36b 37e 38c 39c 40b 41d