Review - Work and Energy

1.

Show that the work done on an object equals (a) the product of the displacement and the component of the force in the direction of the displacement or (b) the product of the force and the component of the displacement in the direction of the force.

2.

A box rests on a horizontal, frictionless surface. A girl pushes on the box with a force of 18 N to the right and a boy pushes on the box with a force of 12 N to the left. The box moves 4.0 m to the right. Find the work done by (a) the girl, (b) the boy, and (c) the net force.

3.

You support an object and move it to the right with a constant velocity. You exert a force F on it (Fig. 1 below) to oppose the gravitational attraction mg of the earth for the object. If you do not raise the object or increase its velocity, there is no increase in the object's potential energy or in its kinetic energy. Do you do work on the object?

4.

An object attached to a string, fixed at one end, lies on a horizontal, frictionless surface. The object is given an initial velocity v and moves in a circle with uniform circular motion. Does the tension in the string do work on the object?

5.

An object of mass m = 2.0 kg is pulled along a surface by a horizontal force F of 12 N to the right a distance s of 4.0 m (Fig. 2 below). The coefficient of friction between the object and the surface is 0.5. Find the work done by (a) F, (b) the normal force F_N,  (c) the weight mg of the object, (d) the frictional force f, (e) the net force.

6.

A raindrop (m = 3.35 x 10^-5 kg) falls vertically at constant speed under the forces of gravity and air resistance. In falling through 100 m, what is the work done by (a) gravity and (b) air resistance?

7.

A 52-kg skier moves down a slope at a speed of 14.0 m/s (31.3 mph). Determine the kinetic energy of the skier.

8.

Show that (a) v ^. v = v² and (b) d(v²) = d(v ^. v) = 2v ^. dv = d(v²) = 2 v dv.

9.

The work-energy theorem states that the work done by the net force on an object of mass m equals the change in kinetic energy of the object. Derive the theorem using (a) a non-calculus approach: Assume a constant F_net acts through a distance x. Write down an expression for the result of a net force and then write the constant acceleration in terms of velocities. (b) a calculus approach:

Integrate     Remember ds = v dt.

10.

In problem #5, if the initial velocity of the object v_o = 0, find the velocity of the object after it has been moved a distance of 8.0 m.

11.

A projectile has a velocity v_A parallel to the earth when at a height h (Fig. 3 below). Find the work done by the gravitational force mg when the projectile goes from A to B.

12.

Use the work-energy theorem to find the speed of the projectile just as it hits the ground at B in Fig. 3 above.

13.

An object, initially at rest, is pulled up an incline that makes an angle of 37^o with the horizontal by a force F = 30 N parallel to the incline. The mass of the object is 2.0 kg and the coefficient of friction between the object and the surface is 0.5. The object moves up the incline a distance s = 4.0 m. Find (a) the frictional force, (b) the work done by F, (c) the work done by the normal force, (d) the work done by the weight of the object, (e) the work done by friction, and (f) the velocity of the object after it has moved 4.0 m.

14.

A 2.0-kg object is pushed along a horizontal surface a distance s = 4.0 m by a force F = 30 N to the right and down at an angle of 37^o to the horizontal. The coefficient of friction between the object and the surface is 0.50. Find (a) the frictional force. (b) the work done by the frictional force (c) the work done by force F, (d) the work done by the normal force F_N, (e) the work done by the weight of the object, (f) the net work done on the object and (g) the speed of the object after it moves through s = 4.0 m if its initial speed = 0.

15.

A constant force of 10 N is exerted to lift a l.0-kg mass a vertical height of 1.0 m. Find (a) the work done by the person, (b) the work done by the gravitational force, (c) the increase in its gravitational energy, and (d) the increase in its kinetic energy. Take g = 10 m/s².

16.

Repeat Problem #15 for a constant force of 12 N.

17.

A box of mass 12 kg slides at a speed of 10 m/s across a smooth level floor, where it enters a rough portion 3.0 m in length. In the rough portion, the box experiences a horizontal frictional force of 72 N. (a) How much work is done by the frictional force? (b) What is the velocity of the box when it leaves the rough surface? (c) What length of rough surface brings the box completely to rest? Take g = 10 m/s².

18.

(a) How much work is required to push a 2.0-kg object up a frictionless inclined plane whose length is 2.0 m and whose height is 1.0 m, if the velocity of the object remains constant? (b) How much work is required to push the object up the plane while increasing its velocity from zero to 3.0 m/s? (c) How much work is required to push the object up the plane at a constant speed if there is a frictional force of 3.0 N between the object and the plane? Take g = 10 m/s².

19.

A block of mass 1.0 kg is placed at the top of an incline of length 125 m and height 62.5 m. The plane has a rough surface. When the block arrives at the bottom of the plane it has a velocity of 25 m/s. What is the magnitude of the constant frictional force acting on the block? Take g = 10m/s^2.

20.

Assume that the total energy of an electron bound to a proton in the hydrogen atom is -21.7 x 10^-19 J. What is the kinetic energy of the electron (we assume the proton is at rest) when the potential energy of the atom is -43.4 x 10^-19 J?

21.

A 2000-kg car is at rest on a level frictionless track. A constant force acts on it for one-half second, after which the car is moving with a speed of 0.20 m/s. Find (a) the magnitude of the force, (b) the kinetic energy of the car, (c) the work done on the car, and (d) the displacement of the car.

22.

Find (d) the work done if the force F = 20 N j and the displacement s = 4 m i and (e) the work done if the force F = (3i + 4j)N and the displacement s = (2i - 2j) m.

23.

An object is subjected to a force along the X-axis as shown in Fig. 4 below. Find the work done on the object as it moves from (a) x = 0 to x = 1 m, (b) x = 1 m to x = 2 m and (c) x = 2 m to x = 3 m. If the mass of the object m = 2.0 kg and its initial velocity v_o = 0, use the work-energy theorem to find (d) v₁ the velocity at x = 1.0 m, (e) v₂ the velocity at x = 2.0 m and (f) v₃ the velocity at x = 3.0 m.

24.

The only force acting on an object is F = (4.0 N + 6.0 N/m x)i. The object, initially at rest, is moved through s = 2.0 mi. Find (a) the work done on the object and (b) its velocity after it has moved 2.0 m.

25.

A spring obeying Hooke's law has k = 50 N/m. When suspended vertically its equilibrium length is 0.50 m. Determine the new equilibrium length when a 0.50 kg object is attached to it. Take g = 10 m/s².

26.

A ball of mass m = 1.0 kg is attached to a spring and the spring is attached to a fixed point P, as shown in Fig. 5 below. The spring cannot bend. The ball is moving in a circle of radius R in a horizontal plane with a velocity v. The spring is massless and the plane on which the ball moves is frictionless. (a) If R = 1.0 m and v = 1.0 m/s, what is the tension in the spring at the point where it attaches to m? (b) If the relaxed length of the spring is 0.90 m, what is the spring constant k? (c) If the ball and spring now rotate with v = 2.0 m/s, what is the new radius of the ball's path. (d) How much work is done on the mass and on the spring.

27.

The potential energy difference U_f - U_i between two points i and f is defined as the negative integral from i to f of F_c ^. ds, where F_c is a conservative force:


Find (a) the potential energy difference and (b) the potential energy for
F_c = -kx² i.

28.

29.

An object of mass m = 2.0 kg is released from rest at the top of the frictionless incline of Fig. 7 below. Taking g = 10 m/s², use energy considerations to find the velocity of the object at the bottom of the incline.

30.

Repeat #29 when µ_k between the object and the plane is 1/4.

31.

When the block in #29 reaches the bottom of the incline it compresses a light spring (Fig. 8 below). If the spring constant k = 480 N/m, what is the maximum compression of the spring?

32.

Figure 9 below is a plot of the potential energy U of a freely falling object as a function of the height y above the ground. E = 16.0 J (shown by the dashed horizontal line) is the total mechanical energy of the object. Find (a) the potential energy and (b) the kinetic energy of the object for y = 0.75 m. Find (c) the mass of the object and (d) v when y = 0.75 m. Take g = 10 m/s².

33.

Figure 10 below is a plot of the potential energy of a mass-spring system. The total mechanical energy E of the system = 0.200 J. Find (a) the potential energy U and (b) the kinetic energy K at x = 0.025 m. Find (c) the spring constant k and (d) the speed of the particle when x = 0.025 given that the mass of the object m = 0.30 kg. Find (e) the amplitude of the motion and (f) the maximum velocity of the object.

34.

A block with mass m = 1.0 kg slides to the right on a horizontal surface. The coefficient of kinetic friction between the surface and the block is 0.3. The block approaches a massless spring with spring constant k = 10 N/m. The dashed vertical line in Fig. 11 below is the unstretched position of the spring. (a) When the block makes contact with the spring at its unstretched position, its velocity is v_ia = 4.0 m/s. It then compresses the spring a distance x and comes to rest. Find x. (b) The block then moves to the left. Find the velocity v_fb when the spring is in its unstretched position. (c) Find the distance d_c from the unstretched position when the block comes to rest. Take g = 10 m/s².

35.

A block of mass m = 2 kg is held at the top of an incline plane that makes an angle of 37^o with the horizontal. The initial position of the block is shown in Fig. 12 (i) below where it is a distance s = 1.0 m from a spring with constant k = 120 N/m. In Fig. 12 (f) below, I show the final position of the block. (a) Look at Fig. 12 (i) and Fig. 12 (f) and decide where to take the zero of gravitational potential energy. (b) Having made your decision in (a) what is the potential energy of the system for the initial position? (c) What is the kinetic energy of the system for the initial position? The coefficient of kinetic friction between the surface and the block is 1/8. (d) What is the frictional force that acts on the block as it goes down the incline? As the block makes contact with the spring, it continues to move until it compresses the spring to a maximum distance x. For the "final position", find (e) the final potential energy, (f) the final kinetic energy, (g) the work done by the frictional force. (h) Put this all together and find x. Take g = 10 m/s².

36.

A small block of mass m slides along the frictionless loop-the-loop shown in Fig. 13 below. Find (a) the minimum height h above the bottom of the track which you must release the block so that it will not leave the track at the top of the hoop and (b) the force of the track on the sphere at P in Fig. 13 when it is released from the height h found in (a). Take g = 10 m/s².

37.

A bob of mass m is suspended from a string of length L. The bob is pulled to the left so the string makes an angle Q with the vertical (Fig. 14 below). The bob is released and vibrates back and forth from position i to c to f, back to c and then back to i, where the motion starts over again. Find the tension in the string at positions i, c, and f.

38.

A point mass m starts with speed v_o at i and moves in a vertical circle along the frictionless surface of radius R (Fig. 15 below). (a) What is the minimum value of v_o such that the mass makes it around the circle. (b) Now take v = 4/5 of the value found in (a).The particle moves up to some point P shown in the figure, loses contact with the track, and travels approximately along the dashed track shown in Fig. 15. Find the position of P. Take g = 10 m/s².

39.

In Fig. 16 below, the first block has a mass m₁ = 2.0 kg and the second block has m₂ = 4.0 kg. The pulley and string are massless. There is no friction in the pulley, but the coefficient of kinetic friction between the first block and the incline is µ_k = 0.55. The blocks are released from rest. Use energy considerations to find the speed of the two blocks when the second block has moved down 2.5 m. Take g = 10 m/s².

40.

A very light rod of length L has a ball of mass m attached to one end. The other end rotates about a pivot without friction. The system rotates in a vertical circle starting at position A in Fig. 17 below with downward velocity v_o. When the ball reaches D, it stops and swings back down in a clockwise direction. Find (a) an expression for v_o in terms of L, m, and g and (b) the tension in the rod at position B. A little sand gets into the pivot and then the ball only reaches C when launched from A. Find (c) the work done by friction during the motion from A to C and (d) how much total work is done by friction when the ball finally comes to rest at B after oscillating back and forth a few times.

41.

A rope with a mass of 0.20 kg and length of 1.0 m hangs, initially at rest, on a frictionless peg that has a negligible radius (Fig. 18 below). Use conservation of energy to find the vertical velocity of the rope just as the end slides off the peg. Take g = 10 m/s².

42.

43.

Given F = 2.0 N/m³ xy² i + 2.0 N/m³ yx² j. Find the work done by this force in going from (a) O to A to C, (b) O to B to C, (c) O to C and (d) 0 to A to C to 0 (Fig. 20 below). (e) Is the force conservative?