Non-Conservative Forces

As you read, pay attention to Figure 7.15 for a comparison of conservative and non-conservative forces. In Figure 7.15 (a), a rock is being "bounced" on an ideal spring with no friction. The mechanical energy does not change, and the rock will continue bouncing indefinitely. In Figure 7.15 (b), the rock is thrown and lands on the ground. When it hits the ground, its kinetic energy is converted to thermal energy and sound. The rock can not "bounce" back up because its mechanical energy is not conserved.

Gravity is a good example of a conservative force we use a lot in physics. Gravitational force is a conservative force because the work gravity does on an object does not depend on the path the object takes. Consequently, gravity is a good candidate to add into the work-energy theorem, where work is only done by gravity: $W=Fd=mad$

Since the acceleration due to gravity is simply $g$ and the direction of motion due to gravity is in the y-axis, we can further build the equation that represents work due to gravity: $W=mg(\Delta y)=\Delta(mgy)$

Previously, we have discovered that work is also equal to the change in kinetic energy (see Section 7.2). So, we can now combine our equation for work due to gravity and our equation for work with respect to the change in kinetic energy: $\Delta(mgy)=\Delta(\frac{1}{2})mv^{2}$ . The $mgy$ in the equation is called the gravitational potential energy. We define potential energy as stored energy due to a system's position: $PE=mgy$ .

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 $W_{\text {net }}=\Delta \mathrm{KE}$ . The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,

$W_{\mathrm{net}}=W_{\mathrm{nc}}+W_{\mathrm{c}}$ ,

so that

$W_{\mathrm{nc}}+W_{\mathrm{c}}=\Delta \mathrm{KE}$ ,

where $W_{\mathrm{nc}}$ is the total work done by all nonconservative forces and $W_{\mathrm{c}}$ is the total work done by all conservative forces.

A person pushing a heavy box up an incline. A force F p applied by the person is shown by a vector pointing up the incline. And frictional force f is shown by a vector pointing down the incline, acting on the box.

Figure 7.16 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 Figure 7.16, 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 $W_{\mathrm{c}}=-\Delta \mathrm{PE}$ . Substituting this equation into the previous one and solving for $W_{\mathrm{nc}}$ gives

$W_{\mathrm{nc}}=\Delta \mathrm{KE}+\Delta \mathrm{PE}$ .

This equation means that the total mechanical energy $(\mathrm{KE}+\mathrm{PE})$ changes by exactly the amount of work done by nonconservative forces. In Figure 7.16, 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 $W_{\mathrm{nc}}=\Delta \mathrm{KE}+\Delta \mathrm{PE}$ to obtain

$\mathrm{KE}_{\mathrm{i}}+\mathrm{PE}_{\mathrm{i}}+W_{\mathrm{nc}}=\mathrm{KE}_{\mathrm{f}}+\mathrm{PE}_{\mathrm{f}}$ .

This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If $W_{\mathrm{nc}}$ is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in Figure 7.16. If $W_{\mathrm{nc}}$ is negative, then mechanical energy is decreased, such as when the rock hits the ground in Figure 7.15(b). If $W_{\mathrm{nc}}$ 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.