Work/Energy with Calculus

1.An object moves according to the function x = t5/2 where x is the distance traveled and t is the time. Its kinetic energy is proportional to

(A) t2 (B) t5/2 (C) t3 (D) t3/2 (E) t° (kinetic energy is constant)

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

(A) Force is the anti-derivative of potential energy(B) Force is the negative gradient of potential energy (C) Potential energy is the negative gradient of force (D) Potential energy is the derivative of force (E) Force is the anti-derivative of potential energy

3.A student throws a ball of mass m upward where the initial potential energy is 0. At a height H the ball has a potential energy of U and is moving upward with a kinetic energy of Ke. Ignoring air resistance, the maximum height achieved by the ball is most nearly

(A) (Ke-U)/mg (B) (U-Ke)/mg (C) (U+Ke)/mg (D) mg/(U-Ke) (E) mg/(U+Ke)

4.A box of mass m is lifted a vertical distance h in time t with a constant velocity. The power supplied by the lifting force is approximately

(A) mght (B) mgh/t (C) zero (D) mgt/h (E) mg/ht

5. A 5 kilogram ball moves 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) = 4x2 3x + 2. The force on the particle at x = 4 meters is

(A) 29 N in +x direction (B) 29N 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 mass m, and the coefficient of sliding friction is represented by µ. The power supplied by the person to the box is

(A) 0 (B) µmg/v (C) µv/mg (D) mg/µv (E) µmgv

7.The force exerted by a spring is given by: F=kx4/2. If k is 100 N/m find the work done from x=0 to x=2.

(A) 100 J (B) 320 J (C) 800 J (D) 1600 J (E) 2400 J

8. A man lifts a mass m at constant speed to a height h in time t. How much work is done by the weight lifter?

(A) mgt (B) zero (C) mgh (D) mgh/t (E) cannot be determined

9. It takes a force F given by F = 20x 12x2 to stretch an irregular spring by a distance x, where F is in newtons and x is in meters. What is the change in potential energy when the spring is stretched 3 meters from its equilibrium position?

(A) 18J (B) 28J (C) 56J (D) 64J (E) 80J

10. On top of a skyscraper of height H, a ball of mass m 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 approximately

(A) mgh-(vo2-vf2) (B) mgh+(vo2+vf2) (C) mgh+(vf2 -vo2) (D) mgh+(vf2 +vo2)

(E)(vf2 -vo2)-mgh

11.A ball attached to a string rotates in a complete circle. The work done during each revolution is

(A) 0 (B) U (C) U+Ke (D) Ke (E) Ke-U

12. The potential energy of two molecules is given by: U=2/r7-4/r5. If r is the distance between two molecules what is the force acting on the particles if r=1m?

(A) 0.75 N (B) 0.67 N (C) 2 N (D) 6 N (E) 10 N

13.A force of 40 N compresses a spring with a spring constant 80 N/m. How much energy is stored in the spring?

(A) 10 J (B) 15 J (C) 20 J (D) 25 J (E) 30 J

14. When the object of mass m1 descends to a distance h, the potential energy of the system has decreases by

(A) (m1 – m2)gh (B) m1gh (C) (m1 + m2)gh (D) ½(ml + m2)gh (E) 0

15. What is the power delivered by gravity to a 6kg block 4 seconds after it has fallen from rest?

(A) 2400 W (B) 1000 W (C) 800 W (D) 1200 W (E) 2000 W

16. If F(x)=2x3-3x2 what is the work done from x=1 to x=2?

(A) 0.5 J (B) 0.8 J (C) 2 J (D) 12 J (E) 0 J

17. A 5 kilogram block is pushed horizontally across a rough surface with a coefficient of kinetic friction of 0.2 by a force F moving at 3 m/s. The work that is done by the force in 20 seconds is

(A) 200 J(B) 400 J(C) 600 J(D) 800 J(E) 1000 J

Questions 18-19

18. The graph represents the force exerted on a particle. What is the work done on the object from x=0 to x= 4 m?

(A) 8 J (B) 10 J (C) 15 J (D) 16 J (E) 20 J

19. What is the work done on the object from x=0 to x=6 m?

(A) 4 J (B) 6 J (C) 10 J (D) 12.5 J (E) 25 J

20.A constant force supplies an average power of 8 watts to a box during a certain time interval. If the box has an average speed of 4 meters per second and the force acts in the same direction as motion of the object, the magnitude of the force is

(A) 8 N (B) 6 N (C) 4 N (D) 2N (E) 1N

Questions 21-23

A ball of mass m is tossed from roof of height h with an initial velocity vo.

21. What is the kinetic energy of the ball at the top of the roof?

(A) 0(B) mgh(C) ½ mvo2(D) mgh-½ mvo2(E) mgh+½ mvo2

22. What is the potential energy of the ball when it hits the ground?

(A) 0(B) mgh/2(C) ½ mvi2(D) mgh/2+½ mvi2 (E) mgh+½ mvi2

23. What is the kinetic energy of the ball right before it strikes the ground?

(A) 0(B) mgh(C) ½ mvi2(D) mgh-½ mvi2(E) mgh+½ mvi2

24.The following graphs represent the net force F as a function of displacement x for an ball that moves in a straight line. Which graph best represents the force that will cause the least change in the kinetic energy of the object from x = 0 to x = x1?

Questions 25-27

A block is attached to a spring that oscillates in simple harmonic motion.

25. At which point is the kinetic energy of the ball at maximum?

(A) Point A(B) Point B(C) Point C(D) Points A&B(E) Kinetic energy is the same at all points

26. What happens to the potential energy as the ball passes from point A to point B?

(A) increases(B) decreases(C) stays the same (D) zero(E) not enough information

27. Which of the following is true about the system?

(A) Potential energy is never equal to Kinetic energy(B) Potential energy is equal to Kinetic energy at all points (C) The system will eventually stop moving (D) There is a restoring force (E) Both B&D

28. A 4 kg mass is moving with a velocity given by v (t) = ¼t4. At t= 1 s, the instantaneous power delivered by the net force is

(A) 1 W (B) 3 W (C) 12 W (D) 14 W (E) 20 W

29. What is the instantaneous power delivered by the net force at the t = 2 s to a 2 kg mass moving according to x(t) = t3/3?

(A) 2 W (B) 10 W (C) 16 W (D) 24 W (E) 32 W

Questions 30 31

A particle of mass m follows the potential energy graph as shown above. The mass is initially at rest at RO.

30. Determine the particle’s speed at position 4ro

(A)(B)(C)(D)(E)

31. The function U(r) = ar-3/2 + 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) (B) (C) (D) (E)

Questions 32-33

The force exerted by a non-uniform spring is F = -kx3. The maximum displacement of the spring is x = 2 m. The spring constant of the spring is 36 N/m and the mass of the block is .5 kg.

32. Find the total energy of the system.

(A) 12 J (A) 144 J (C) 36 J (D) 120 J (E) 432 J

33. Find the maximum speed of the system.

(A) 9.8 m/s (B) 24 m/s (C) 15 m/s (D) 20 m/s (E) 27 m/s

Question 34-35

A block of mass m slides down an incline and experiences a friction force with a coefficient of kinetic friction µ on the ramp and the track it follows. The block comes to rest at point 4.

34. At which point is the kinetic energy the greatest?

(A) Point 1(B) Point 2(C) Point 3(D) Points 4(E) Between Point 1&2

35. Which of the following best represents the energy lost to friction from point 1 to point 4?

(A) µmgH(B) MgH(C) (D) 2mg/H(E) 2gH

36. A mass moves under the influence of a potential energy given by: U(x) = x3 – 2x2

At x=2, the force on the mass 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 mass is attached to a spring on a frictionless horizontal surface. It is displaced by an amount of x meters and then released. Which of the following statements are true?

PE and KE will never be equal.
KE is largest when the mass is passing through the equilibrium point.
PE is largest when the mass has a displacement 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=9r2+4. Find the acceleration of the object at r=2

(A) 1 m/s2 (B) 4 m/s2 (C) 6 m/s2 (D) 9 m/s2 (E) 12 m/s2

39. A restoring force F = -2x3 acts on an object, where x is the displacement of the object from its equilibrium position. How much work must be done to move the object from x= 0 to x= 2 m?

A) 2 J B) 24 J C) 8 J D) 10 J E) 15 J

Questions 40-41

F(x) = sin x

40. Find the work done from x = 0 to x = π/2.

(A) 0 J (B) 1 J (C) π/2 J (D) 5 J (E) 10 J

41. Find the work done from x = 0 to x = π.

(A) 0 J (B) 1 J (C) 1.5 J (D) -1 J (E) 4 J

Answers 1c 2b 3c 4b 5b 6e 7b 8c 9a 10c 11a 12d 13a 14a 15d 16a 17c 18a 19b 20d 21c 22a 23e 24c 25b 26b 27d 28a 29e 30d 31a 32b 33b 34b 35b 36b 37e 38c 39c 40b 41d