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: \( PE=mgy \).
Applying Energy Conservation with Nonconservative Forces
When no change in potential energy occurs, applying \(\mathrm{KE}_{\mathrm{i}}+\mathrm{PE}_{\mathrm{i}}+W_{\mathrm{nc}}=\mathrm{KE}_{\mathrm{f}}+\mathrm{PE}_{\mathrm{f}}\) amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation \(\mathrm{KE}_{\mathrm{i}}+\mathrm{PE}_{\mathrm{i}}+W_{\mathrm{nc}}=\mathrm{KE}_{\mathrm{f}}+\mathrm{PE}_{\mathrm{f}}\) says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.