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10-7 NEWTON’S SECOND LAW FOR ROTATION 279

10-7NEWTON’S SECOND LAW FOR ROTATION

After reading this module, you should be able to...

10.28Apply Newton’s second law for rotation to relate the
net torque on a body to the body’s rotational inertia and

rotational acceleration, all calculated relative to a specified

rotation axis.

●The rotational analog of Newton’s second law is
tnetIa,

wheretnetis the net torque acting on a particle or rigid body,

Iis the rotational inertia of the particle or body about the

rotation axis, and ais the resulting angular acceleration about

that axis.

Learning Objective

Key Idea

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 tneton a rigid body to the angular
accelerationathat torque causes about a rotation axis. We do so by analogy with
Newton’s second law (Fnetma) for the acceleration aof a body of mass mdue
to a net force Fnetalong a coordinate axis. We replace Fnetwithtnet,mwithI, and a
withain radian measure, writing

tnetIa (Newton’s second law for rotation). (10-42)

Proof of Equation 10-42

We prove Eq. 10-42 by first considering the simple situation shown in Fig. 10-17.
The rigid body there consists of a particle of mass mon one end of a massless rod
of length r. The rod can move only by rotating about its other end, around a rota-
tion 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 Ftof the force (the
component that is tangent to the circular path) can accelerate the particle along
the path. We can relate Ftto the particle’s tangential acceleration atalong the
path with Newton’s second law, writing

Ftmat.

The torque acting on the particle is, from Eq. 10-40,

tFtrmatr.

From Eq. 10-22 (atar) we can write this as

tm(ar)r(mr^2 )a. (10-43)

The quantity in parentheses on the right is the rotational inertia of the particle
about the rotation axis (see Eq. 10-33, but here we have only a single particle).
Thus, using Ifor the rotational inertia, Eq. 10-43 reduces to

tIa (radian measure). (10-44)

If more than one force is applied to the particle, Eq. 10-44 becomes

tnetIa (radian measure), (10-45)

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.

F

:

Figure 10-17A simple rigid body, free to

rotate about an axis through O, consists of

a particle of mass mfastened to the end of

a rod of length rand negligible mass. An

applied force Fcauses the body to rotate.

:

O

x

y

Rod

θ

Rotation axis

r

m

Fr

Ft

φ

F

The torque due to the tangential

component of the force causes

an angular acceleration around

the rotation axis.