Page 130 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 10] ROTATIONAL MOTION 115

and in this time it will have turned through an angular displacement of
1
θ = ω 0 t + αt 2
2
A relationship that does not involve the time t directly is sometimes useful:
2
2
ω = ω + 2αθ
t 0
SOLVED PROBLEM 10.7
An engine requires 5 s to go from its idling speed of 600 rev/min to 1200 rev/min. (a) What is its angular
acceleration? (b) How many revolutions does it make in this period?
(a) The initial and final angular velocities of the engine are, respectively,
2π rad/rev

ω 0 = (600 rev/min) = 63 rad/s
60 s/min

2π rad/rev
ω f = (1200 rev/min) = 126 rad/s
60 s/min
and so its angular acceleration is
ω f − ω 0 126 rad/s − 63 rad/s 2
α = = = 12.6 rad/s
t 5s
(b) The angle through which the engine turns is
1
2
1
2
θ = ω 0 t + αt = (63 rad/s)(5s) + ( )(12.6 rad/s )(5s) 2
2 2
= 472.5 rad
Since there are 2π rad in 1 rev,
472.5 rad
θ = = 75.2rev
2π rad/rev
SOLVED PROBLEM 10.8

A phonograph turntable initially rotating at 3.5 rad/s makes three complete turns before coming to a stop.
(a) What is its angular acceleration? (b) How much time does it take to come to a stop?

(a) The angle in radians that corresponds to 3 rev is
θ = (3rev)(2π rad/rev) = 6π rad
2
2
From the formula ω = ω + 2αθ we find
0
2
ω − ω 2 0 0 − (3.5 rad/s) 2
f
α = = =−0.325 rad/s 2
2θ (2)(6π rad)
(b) Since ω f = ω 0 + αt, we have here
ω f − ω 0 0 − 3.5 rad/s
t = = 2 = 10.8s
α −0.325 rad/s
MOMENT OF INERTIA
The rotational analog of mass is a quantity called moment of inertia. The greater the moment of inertia of a body,
the greater its resistance to a change in its angular velocity. The value of the moment of inertia I of a body about
a particular axis of rotation depends not only upon the body’s mass but also upon how the mass is distributed
about the axis.