6.10

Concepts & Calculations

This section contains examples that discuss one or more conceptual questions, followed by a related quantitative problem. Example 15 reviews the important concept of work and illustrates how forces can give rise to positive, negative, and zero work. Example16 examines the all-important conservation of mechanical energy and the work–energy theorem.

| Concepts & Calculations Example 15 | | | Skateboarding and Work

The skateboarder in Figure 6-24a is coasting down a ramp, and there are three forces acting on her: her weight , a frictional force that opposes her motion, and a normal force . Determine the net work done by the three forces when she coasts for a distance of 9.2 m.

Figure 6-24 (a) Three forces act on the skateboarder: , , . For clarity, the frictional force is not drawn to scale. (b) The orientation of the three forces relative to the displacement of the skateboarder.

Concept Questions and Answers Figure 6-24b shows each force, along with the displacement of the skateboarder. By examining these diagrams and without doing any numerical calculations, determine whether the work done by each force is positive, negative, or zero. Provide a reason for each answer.

Answer The work done by a force is positive if the force has a component that points in the same direction as the displacement. The work is negative if there is a force component pointing opposite to the displacement. The work done by the weight is positive, because the weight has a component that points in the same direction as the displacement. The top drawing in Figure 6-24b shows this weight component. The work done by the frictional force is negative, because it points opposite to the direction of the displacement ( in the middle drawing). The work done by the normal force is zero, because the normal force is perpendicular to the displacement ( in the bottom drawing) and does not have a component along the displacement.

Solution The work W done by a force is given by Equation 6.1 as , where F is the magnitude of the force, s is the magnitude of the displacement, and is the angle between the force and the displacement. Figure 6-24b shows this angle for each of the three forces. The work done by each force is computed in the table below. The net work is the algebraic sum of the three values.

Force | Magnitude F of the Force | Angle | Magnitude s of the Desplacement | Work Done by the Force
| 675 N | 65.0° | 9.2 m |
| 125 N | 180.0° | 9.2 m |
| 612 N | 90.0° | 9.2 m |

The net work done by the three forces is

| Cecepts & Calculations Example 16 | | | Conservation of Mechanical Energy and the Work–Energy Theorem

Figure 6-25 shows a 0.41-kg block sliding from A to B along a frictionless surface. When the block reaches B, it continues to slide along the horizontal surface BC where a kinetic frictional force acts. As a result, the block slows down, coming to rest at C. The kinetic energy of the block at A is 37 J, and the heights of A and B are 12.0 and 7.0 m above the ground, respectively. (a) What is the kinetic energy of the block when it reaches B? (b) How much work does the kinetic frictional force do during the BC segment of the trip?

Figure 6-25 The block slides on a frictionless surface from A to B. From B to C, a kinetic frictional force slows down the block until it comes to rest at C.

Concept Questions and Answers Is the total mechanical energy of the block conserved as the block goes from A to B? Why or why not?

Answer The total mechanical energy of the block is conserved if we can show that the net work done by the nonconservative forces is zero, or (see Section 6.5). Only two forces act on the block during its trip from A to B: its weight and the normal force. The block’s weight, as we have seen in Section 6.4, is a conservative force. The normal force, on the other hand, is a nonconservative force. However, it is always perpendicular to the displacement of the block, so it does no work. Thus, we conclude that , with the result that the total mechanical energy is conserved during the AB part of the trip.

When the block reaches point B, has its kinetic energy increased, decreased, or remained the same relative to what it had at A? Provide a reason for your answer.

Answer As we have seen, the total mechanical energy is the sum of the kinetic and gravitational potential energies and remains constant during the trip. Therefore, as one type of energy decreases, the other must increase for the sum to remain constant. Since B is lower than A, the gravitational potential energy at B is less than that at A. As a result, the kinetic energy at B must be greater than that at A.

Is the total mechanical energy of the block conserved as the block goes from B to C? Justify your answer.

Answer During this part of the trip, a kinetic frictional force acts on the block. This force is nonconservative, and it does work on the block, just as the frictional force does work on the skateboarder in Concepts & Calculations Example 15. Consequently, the net work Wnc done by nonconservative forces is not zero , so the total mechanical energy is not conserved during this part of the trip.

Solution

a. | Since the total mechanical energy is conserved during the AB segment, we can set the total mechanical energy at B equal to that at A.

Solving this equation for the kinetic energy at B gives

As expected, the kinetic energy at B is greater than that at A.
b. | During the BC part of the trip, the total mechanical energy is not conserved because a kinetic frictional force is present. The work Wnc done by this nonconservative force is given by the work–energy theorem (see Equation 6.8) as the final total mechanical energy minus the initial total mechanical energy:

Rearranging this equation gives

In the preceding equation we have noted that the kinetic energy KEC at C is equal to zero, because the block comes to rest at this point. The term is also zero, because the two heights are the same. Thus, the work Wnc done by the kinetic frictional force during the BC part of the trip is

The work done by the nonconservative frictional force is negative, because this force points opposite to the displacement of the block.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.