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 Nonconservative Forces Affect Mechanical Energy

Mechanical energy may not be conserved when nonconservative forces act. For example, when a car is brought to a stop by friction on level ground, it loses kinetic energy, which is dissipated as thermal energy, reducing its mechanical energy. Figure 7.15 compares the effects of conservative and nonconservative forces. We often choose to understand simpler systems such as that described in Figure 7.15(a) first before studying more complicated systems as in Figure 7.15(b).

(a) A system is shown in three situations. First, a rock is dropped onto a spring attached to the ground. The rock has potential energy P E sub 0 at the highest point before it is dropped on the spring. In the second situation, the rock has fallen onto the spring and the spring is compressed and has potential energy P E sub s. And in the third situation, the spring pushes the rock into the air; then the rock has some kinetic and some potential energy, labeled as K E plus P E sub g prime. (b) A rock is at some height above the ground, having potential energy P E sub g, and as it hits the ground all of the rock’s energy is used to produce heat, sound, and deformation of the ground.

Figure 7.15 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.