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378 CHAPTER 12 Rotation of a Rigid Body

Evaluating this expression at t 3.0 s, we find
2 3
(3.0s) (3.0 s)
2
f f 60 radians>s a b 180 radians
0
2 18 s
Hence the number of revolutions during the acceleration is
f f 0 180 radians
[number of revolutions] 29 revolutions (12.25)
2p 2p

As discussed in Section 2.7, similar integration techniques can be applied to deter-
mine any component of the translational velocity and the position when the time-
dependent net force and, thus, the time-dependent translational acceleration are known.
In Section 2.7 we also examined the case when the acceleration is a known function
of the velocity; in that case, integration provides t as a function of v (and v ), which can
0
sometimes be inverted to find v as a function of t.
We saw in Chapters 7–9 that a conservation-of-energy approach is often the eas-
iest way to determine the motion when the forces are known as a function of position.
Now we have seen that direct integration of the equations of motion can be applied when
the translational or angular acceleration is known as a function of time or of velocity.

✔ Checkup 12.4

QUESTION 1: Beginning from rest at t 0, the angular velocity of a merry-go-round
increases in proportion to the square root of the time t. By what factor is the angular
position of the merry-go-round at t 4 s greater than it was at t 1 s?
QUESTION 2: A car on a circular roadway accelerates from rest beginning at t 0, so
that its angular acceleration increases in proportion to the time t. With what power of
time does its centripetal acceleration increase?
2 3 4 5
(A) t (B) t (C) t (D) t (E) t

12.5 KINETIC ENERGY OF ROTATION;
MOMENT OF INERTIA

A rigid body is a system of particles, and as for any system of particles, the total kinetic
energy of a rotating rigid body is simply the sum of the individual kinetic energies of
all the particles (see Section 10.4). If the particles in the rigid body have masses m , m ,
1 2
m ,... and speeds v ,v ,v ,..., then the kinetic energy is
3 1 2 3
2
2
2
1
1
1
K m v m v m v (12.26)
2 2
2
3 3
2
2
1 1
In a rigid body rotating about a given axis, all the particles move with the same angu-
lar velocity along circular paths. By Eq. (12.11), the speeds of the particles along
their paths are proportional to their radial distances:
v R , v R , v R , (12.27)
1
3
3
1
2
2
and hence the total kinetic energy is
1
1
1
2 2
2
2
2 2
K m R m R m R
3
3
2
2
1 1
2
2
2