How the Work-Energy Theorem Applies

How the Work-Energy Theorem Applies

Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the mechanical energy of a system. As noted in Kinetic Energy and the Work-Energy Theorem, the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or \({W}_{\text{net}}=\text{ΔKE}\). The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,

so that

where \({W}_{\text{nc}}\) is the total work done by all nonconservative forces and \({W}_{\text{c}}\) is the total work done by all conservative forces.

A person pushes a crate up a ramp, doing work on the crate. Friction and gravitational force (not shown) also do work on the crate; both forces oppose the person’s push. As the crate is pushed up the ramp, it gains mechanical energy, implying that the work done by the person is greater than the work done by friction.

Consider the figure above, in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that \({W}_{\text{c}}=-\text{Δ}\text{PE}\). Substituting this equation into the previous one and solving for \({W}_{\text{nc}}\) gives

This equation means that the total mechanical energy \(\left(\text{KE + PE}\right)\) changes by exactly the amount of work done by nonconservative forces. In the figure above, this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.

We rearrange \({W}_{\text{nc}}=\text{Δ}\text{KE}+\text{Δ}\text{PE}\) to obtain

This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If \({W}_{\text{nc}}\) is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in the figure above. If \({W}_{\text{nc}}\) is negative, then mechanical energy is decreased, such as when the rock hits the ground in figure (b) below from the previous lesson. If \({W}_{\text{nc}}\) is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.

Comparison of the effects of conservative and nonconservative forces on the mechanical energy of a system. (a) A system with only conservative forces. When a rock is dropped onto a spring, its mechanical energy remains constant (neglecting air resistance) because the force in the spring is conservative. The spring can propel the rock back to its original height, where it once again has only potential energy due to gravity. (b) A system with nonconservative forces. When the same rock is dropped onto the ground, it is stopped by nonconservative forces that dissipate its mechanical energy as thermal energy, sound, and surface distortion. The rock has lost mechanical energy.