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400 CHAPTER 13 Dynamics of a Rigid Body

y According to this equation, the torque provided by a force of a given magnitude F is
maximum if the force is at right angles to the radius ( 90 ), and it is zero if the
force is parallel to the radius ( 0 or 180 ).
The quantity R sin appearing in Eq. (13.14) has a simple geometric interpreta-

R
tion: it is the perpendicular distance between the line of action of the force and the
F axis of rotation (see Fig. 13.5); this perpendicular distance is called the moment arm
x
O of the force. Hence, Eq. (13.14) states that the torque equals the magnitude of the
R sin is force multiplied by the moment arm.
moment arm.
To find a quantitative relationship between torque and angular acceleration, we
rotation axis
recall from Eq. (13.6) that the power delivered by a torque acting on a body is
FIGURE 13.5 The distance between the
center of rotation and the point of applica- dW
t (13.15)
tion of the force is R. The perpendicular dt
distance between the center of rotation and
The work–energy theorem tells us that the work dW equals the change of kinetic
the line of action of the force is R sin .
energy in the small time interval dt. The small change in the kinetic energy
2
1
1
is dK d ( I ) I 2 d I d . Thus,
2
2
dW I d (13.16)
Inserting this into the left side of Eq. (13.15), we find
I d
t (13.17)
dt
Canceling the factor of on both sides of the equation, we obtain

d
I t (13.18)
dt
But d dt is the angular acceleration ; hence

equation of rotational motion I t (13.19)

This is the equation for rotational motion. As we might have expected, this equation
says that the angular acceleration is directly proportional to the torque. Equation (13.19)
is mathematically analogous to Newton’s Second Law, ma F, for the translational
motion of a particle; the moment of inertia takes the place of the mass, the angular
acceleration the place of the acceleration, and the torque the place of the force.
In our derivation of Eq. (13.19) we assumed that only one external force is acting
on the rigid body. If several forces act, then each produces its own torque. If an indi-
vidual torque would produce an angular acceleration in the rotational direction chosen
as positive, it is reckoned as positive, and if a torque would produce an angular accel-
eration in the opposite direction, it is reckoned as negative.The net torque is the sum
of these individual torques, and the angular acceleration is proportional to this net
torque:

equation of rotational motion I t (13.20)
for net torque net

In the evaluation of the net torque, we need to take into account all the external forces
acting on the rigid body, but we can ignore the internal forces that particles in the
body exert on other particles also in the body.The torques of such internal forces cancel
(this is an instance of the general result mentioned in Section 10.4: for a rigid body, the
work of internal forces cancels).