Newton's Second Law for Rotation

A torque can cause rotation of a rigid body, as when you use a torque to rotate a door. Here we want to relate the net torque on a rigid body to the angular acceleration it causes about a rotation axis. We do so by analogy with Newton's second law ( ) for the acceleration a of a body of mass due to a net force along a coordinate axis. We replace with , with , and with , writing

(Newton's second law for rotation), 11-34

where must be in radian measure.

Proof of Equation 11-34

We prove Eq. 11-34 by first considering the simple situation shown in Fig. 11-16.

The rigid body there consists of a particle of mass on one end of a massless rod of length . The rod can move only by rotating about its other end, around a rotation axis (an axle) that is perpendicular to the plane of the page. Thus, the particle can move only in a circular path that has the rotation axis at its center.

A force acts on the particle. However, because the particle can move only along the circular path, only the tangential component of the force (the component that is tangent to the circular path) can accelerate the particle along the path. We can relate to the particle's tangential acceleration along the path with Newton's second law, writing

.

The torque acting on the particle is, from Eq. 11-32,

.

From Eq. 11-22 ( ) we can write this as

11-35

The quantity in parentheses on the right side of Eq. 11-35 is the rotational inertia of the particle about the rotation axis (see Eq. 11-26). Thus, Eq. 11-35 reduces to

For the situation in which more than one force is applied to the particle, we can generalize Eq. 11-26 as

which we set out to prove. We can extend this equation to any rigid body rotating about a fixed axis, because any such body can always be analyzed as an assembly of single particles.

11-10 Work and Rotational Kinetic Energy

As we discussed in Chapter 7, when a force causes a rigid body of mass to accelerate along a coordinate axis, it does work on the body. Thus, the body's kinetic energy ( ) can change. Suppose it is the only energy of the body that changes. Then we relate the change in kinetic energy to the work with the work-kinetic energy theorem (Eq. 7-10), writing

(work-kinetic energy theorem). (11-41)

For motion confined to an axis, we can calculate the work with Eq. 7-32,

This reduces to when is constant and the body's displacement is d.

The rate at which the work is done is the power, which we can find with Eqs. 7-43

(power, one-dimensional motion). (11-43)

Now let us consider a rotational situation that is similar. When a torque accelerates a rigid body in rotation about a fixed axis, it does work on the body. Therefore, the body's rotational kinetic energy ( ) can change. Suppose that it is the only energy of the body that changes. Then we can still relate the change in kinetic energy to the work with the work-kinetic energy theorem, except now the kinetic energy is a rotational kinetic energy:

(work-kinetic energy theorem). (11-44)

Here, is the rotational inertia of the body about the fixed axis and and are the angular speeds of the body before and after the work is done, respectively. Also, we can calculate the work with a rotational equivalent of Eq. 11-42,

(work, rotation about fixed axis),(11-45)

where is the torque doing the work , and and are the body's angular positions before and after the work is done, respectively. When is constant, Eq. 11-45 reduces to

(work, constant torque). (11-46)

The rate at which the work is done is the power, which we can find with the rotational equivalent of Eq. 11-43,

power, rotation about fixed axis). (11-47)