6.6

Nonconservative Forces and the Work–Energy Theorem

Most moving objects experience nonconservative forces, such as friction, air resistance, and propulsive forces, and the work Wnc done by the net external nonconservative force is not zero. In these situations, the difference between the final and initial total mechanical energies is equal Wnc, to according to (Equation 6.8). Consequently, the total mechanical energy is not conserved. The next two examples illustrate how Equation 6.8 is used when nonconservative forces are present and do work.

| Example 11 | | | The Steel Dragon, Revisited

In Example 10, we ignored nonconservative forces, such as friction. In reality, however, such forces are present when the roller coaster descends. The actual speed of the riders at the bottom is 41.0 m /s, which is less than that determined in Example 10. Assuming again that the coaster has a speed of 3.0 m /s at the top, find the work done by nonconservative forces on a 55.0-kg rider during the descent from a height h0 to a height hf , where .
Reasoning Since the speed at the top, the final speed, and the vertical drop are given, we can determine the initial and final total mechanical energies of the rider. The work–energy theorem, , can then be used to determine the work Wnc done by the nonconservative forces.
Solution The work–energy theorem is

| (6.8)

Rearranging the terms on the right side of this equation gives

Problem solving insight As illustrated here and in Example 3, a nonconservative force such as friction can do negative or positive work. It does negative work when it has a component opposite to the displacement and slows down the object. It does positive work when it has a component in the direction of the displacement and speeds up the object.

| Example 12 | | | Fireworks

A 0.20-kg rocket in a fireworks display is launched from rest and follows an erratic flight path to reach the point P, as Figure 6-20 shows. Point P is 29 m above the starting point. In the process, 425 J of work is done on the rocket by the nonconservative force generated by the burning propellant. Ignoring air resistance and the mass lost due to the burning propellant, find the speed vf of the rocket at the point P.

Figure 6-20 A fireworks rocket, moving along an erratic flight path, reaches a point P that is 29 m above the launch point.

Reasoning The only nonconservative force acting on the rocket is the force generated by the burning propellant, and the work done by this force is . Because work is done by a nonconservative force, we use the work–energy theorem in the form to find the final speed vf of the rocket.
Solution From the work–energy theorem we have

| (6.8)

Solving this expression for the final speed of the rocket and noting that the initial speed of the rocket at rest is , we get

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