9.5

Rotational Work and Energy

Work and energy are among the most fundamental and useful concepts in physics. Chapter 6 discusses their application to translational motion. These concepts are equally useful for rotational motion, provided they are expressed in terms of angular variables.
The work W done by a constant force that points in the same direction as the displacement is W = Fs (Equation 6.1), where F and s are the magnitudes of the force and displacement, respectively. To see how this expression can be rewritten using angular variables, consider Figure 9-22. Here a rope is wrapped around a wheel and is under a constant tension F. If the rope is pulled out a distance s, the wheel rotates through an angle (Equation 8.1), where r is the radius of the wheel and is in radians. Thus, , and the work done by the tension force in turning the wheel is . But Fr is the torque applied to the wheel by the tension, so the rotational work can be written as follows:

| Definition of Rotational Work
| The rotational work WR done by a constant torque in turning an object through an angle is

| (9.8)

Figure 9-22 The force does work in rotating the wheel through the angle .

Requirement: must be expressed in radians.
SI Unit of Rotational Work: joule (J)

6.2 discusses the work–energy theorem and kinetic energy. There we saw that the work done on an object by a net external force causes the translational kinetic energy () of the object to change. In an analogous manner, the rotational work done by a net external torque causes the rotational kinetic energy to change. A rotating body possesses kinetic energy, because its constituent particles are moving. If the body is rotating with an angular speed , the tangential speed vT of a particle at a distance r from the axis is (Equation 8.9). Figure 9-23 shows two such particles. If a particle’s mass is m, its kinetic energy is . The kinetic energy of the entire rotating body, then, is the sum of the kinetic energies of the particles:

In this result, the angular speed is the same for all particles in a rigid body and, therefore, has been factored outside the summation. According to Equation 9.6, the term in parentheses is the moment of inertia, , so the rotational kinetic energy takes the

| Definition of Rotational Kinetic Energy
| The rotational kinetic energy KER of a rigid object rotating with an angular speed about a fixed axis and having a moment of inertia I is

| (9.9)

Figure 9-23 The rotating wheel is composed of many particles, two of which are shown.

Requirement: must be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)

Need more practice?

Interactive LearningWare9.3
A car is moving with a speed of 27.0 m/s. Each wheel has a radius of 0.300 m and a moment of inertia of 0.850 kg · m2. The car has a total mass (including the wheels) of 1.20 × 103 kg. Find the total kinetic energy of the car, translational plus rotational.

following form:

CONCEPTS AT A GLANCE Kinetic energy is one part of an object’s total mechanical energy. The total mechanical energy is the sum of the kinetic and potential energies and obeys the principle of conservation of mechanical energy (see 6.5). The Concepts-at-a-Glance chart in Figure 9-24 shows how rotational kinetic energy is incorporated into this principle. Specifically, we need to remember that translational and rotational motion can occur simultaneously. When a bicycle coasts down a hill, for instance, its tires are both translating and rotating. An object such as a rolling bicycle tire has both translational and rotational kinetic energies, so that the total mechanical energy is

Here m is the mass of the object, v is the translational speed of its center of mass, I is its moment of inertia about an axis through the center of mass, is its angular speed, and h is the height of the object’s center of mass relative to an arbitrary zero level. Mechanical energy is conserved if Wnc, the net work done by external nonconservative forces and external torques, is zero. If the total mechanical energy is conserved as an object moves, its final total mechanical energy Ef equals its initial total mechanical energy E0:Ef = E0.

Figure 9-24 CONCEPTS AT A GLANCE The principle of conservation of mechanical energy can be applied to a rigid object that has both translational and rotational motion, provided the rotational kinetic energy is included in the total mechanical energy and provided that Wnc, the net work done by external nonconservative forces and torques, is zero. The principle can be applied to the motion of this mountain biker as he performs a flip high in the air, to the extent that the shapes of his bike and his body are constant and air resistance is negligible. (Chris McGrath/Getty Images Sport Services)

Example 13, like Interactive Learning Ware 9.3, also illustrates the effect of combined translational and rotational motion, but in the context of how the total mechanical energy of a cylinder is conserved as it rolls down an incline.

| Example 13 | | | Rolling Cylinders

A thin-walled hollow cylinder (mass = mh, radius = rh) and a solid cylinder (mass = ms, radius = rs) start from rest at the top of an incline (Figure 9-25). Both cylinders start at the same vertical height h0 and roll down the incline without slipping. All heights are measured relative to an arbitrarily chosen zero level that passes through the center of mass of a cylinder when it is at the bottom of the incline (see the drawing). Ignoring energy losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom.

Figure 9-25 A hollow cylinder and a solid cylinder start from rest and roll down the incline plane. The conservation of mechanical energy can be used to show that the solid cylinder, having the greater translational speed, reaches the bottom first.

Reasoning Only the conservative force of gravity does work on the cylinders, so the total mechanical energy is conserved as they roll down. The total mechanical energy E at any height h above the zero level is the sum of the translational kinetic energy () the rotational kinetic energy (), and the gravitational potential energy (mgh):

As the cylinders roll down, potential energy is converted into kinetic energy, but the kinetic energy is shared between the translational form () and the rotational form (). The object with more of its kinetic energy in the translational form will have the greater translational speed at the bottom of the incline. We expect the solid cylinder to have the greater translational speed, because more of its mass is located near the rotational axis and, thus, possesses less rotational kinetic energy.
Solution The total mechanical energy Ef at the bottom (hf = 0 m) is the same as the total mechanical energy E0 at the top ():

Since each cylinder rolls without slipping, the final rotational speed and the final translational speed vf of its center of mass are related according to Equation 8.12, , where r is the radius of the cylinder. Substituting this expression for into the energy-conservation equation and solving for vf yields

Setting and for the hollow cylinder and then setting , and for the solid cylinder (see Table 9-1), we find that the two cylinders have the following translational speeds at the bottom of the incline:
Hollow cylinder

Solid cylinder

Check Your Understanding 4

Two solid balls are placed side by side at the top of an incline plane and, starting from rest, are allowed to roll down the incline. Which ball, if either, has the greater translational speed at the bottom if (a) they have the same radii but one is more massive than the other, and (b) they have the same mass but one has a larger radius? (The answers are given at the end of the book.)
Background: The conservation of total mechanical energy is the central concept here. Note that the mass of an object is not the same as its moment of inertia.
For similar questions (including calculational counterparts), consult Self-Assessment Test 9.2. This test is described at the end of Section 9.6.

The solid cylinder, having the greater translational speed, arrives at the bottom first.

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