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: $PE=mgy$ .

Example 7.9 Calculating Distance Traveled: How Far a Baseball Player Slides

Consider the situation shown in Figure 7.17, where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.

A baseball player slides to stop in a distance d. the displacement d is shown by a vector towards the left and frictional force f on the player is shown by a small vector pointing towards the right equal to four hundred and fifty newtons. K E is equal to half m v squared, which is equal to f times d.

Figure 7.17 The baseball player slides to a stop in a distance d . In the process, friction removes the player's kinetic energy by doing an amount of work fd equal to the initial kinetic energy.

Strategy

Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the work-energy theorem, the work done by friction, which is negative, is added to the initial kinetic energy to reduce it to zero. The work done by friction is negative, because $\mathbf{f}$ is in the opposite direction of the motion (that is, $\theta=180^{\circ} \text {, and so } \cos \theta=-1$ . Thus $W_{\text {nc }}=-f d \text {. }$ . Thus $W_{\mathrm{nc}}=-f d$ . The equation simplifies to

$\frac{1}{2} m v_{\mathrm{i}}^{2}-f d=0$

$f d=\frac{1}{2} m v_{\mathrm{i}}^{2}$

This equation can now be solved for the distance $d$ .

Solution

Solving the previous equation for $d$ and substituting known values yields

$\begin{aligned} d &=\frac{m v_{\mathrm{i}}^{2}}{2 f} \\ &=\frac{(65.0 \mathrm{~kg})(6.00 \mathrm{~m} / \mathrm{s})^{2}}{(2)(450 \mathrm{~N})} \\ &=2.60 \mathrm{~m} \end{aligned}$

Discussion

The most important point of this example is that the amount of nonconservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large mechanical energy, than to stop a mosquito.