There are three common situations where no work is done. The most obvious is when the object doesn’t move $(\text{Δ}\vec{r}=\vec{0})$. If you were to hold a 200 lb weight over your head, you might break out in a sweat and your arms would tire. You might feel that you had done a lot of work, but you would have done zero work in the physics sense because the weight was not displaced and thus you transferred no energy to it. A force acting on a particle does no work unless the particle is displaced.

FIGURE 9.14 shows a particle moving in uniform circular motion. As you learned in Chapter 8, uniform circular motion requires a force pointing toward the center of the circle. How much work does this force do?

FIGURE 9.14 A perpendicular force does no work.

Zero! Tactics Box 9.1 showed that a force perpendicular to the displacement does no work. Circular motion extends this idea to motion along a curve. A particle in uniform circular motion has a constant speed, so there’s no change in kinetic energy and hence no energy is transferred into or out of the system. Thus a force everywhere perpendicular to the motion does no work.

Last, consider the roller skater in FIGURE 9.15 who straightens her arms and pushes off from a wall. She applies a force to the wall and thus, by Newton’s third law, the wall applies a force $\vec{F}{}_{\text{W on S}}$ to her. How much work does this force do?

FIGURE 9.15 Does the wall do work on the skater?

Surprisingly, zero. The reason is subtle but worth discussing because it gives us insight into how energy is transferred and transformed. The skater differs from suitcases and rockets in two important ways. First, the skater, as she extends her arms, is a deformable object. We cannot use the particle model for a deformable object. Second, the skater has an internal source of energy. Because she’s a living object, she has an internal store of chemical energy that is available through metabolic processes.

Although the skater’s center of mass is displaced, the palms of her hands—where the force is exerted—are not. The particles on which force $F_{\text{W on S}}$ acts have no displacement, and we’ve just seen that there’s no work without displacement. The force acts, but the force doesn’t push any physical thing through a displacement. Hence no work is done.

But the skater indisputably gains kinetic energy. How? Recall, from the energy overview that started this chapter, that the full energy principle is $\text{Δ}{E}_{\text{sys}}=W_{\text{ext}}$. A system can gain kinetic energy without any work being done if it can transform some other energy into kinetic energy. In this case, the skater transforms chemical energy into kinetic energy. The same is true if you jump straight up from the ground. The ground applies an upward force to your feet, but that force does no work because the point of application—the soles of your feet—has no displacement while you’re jumping. Instead, your increased kinetic energy comes via a decrease in your body’s chemical energy. A brick cannot jump or push off from a wall because it cannot deform and has no usable source of internal energy.