c/Analogies between rota-
tional and linear quantities.
d/We construct a coordi-
nate system that coincides with
the location and motion of the
moving point of interest at a
certain moment.
In the absence of any torque, a rigid body will rotate indefinitely
with the same angular velocity. If the angular velocity is changing
because of a torque, we define an angular acceleration,α=
dω
dt
, [definition of angular acceleration]The symbol is the Greek letter alpha. The units of this quantity are
rad/s^2 , or simply s−^2.
The mathematical relationship betweenωandθis the same as
the one betweenvandx, and similarly forαanda. We can thus
make a system of analogies, c, and recycle all the familiar kinematic
equations for constant-acceleration motion.
The synodic period example 12
Mars takes nearly twice as long as the Earth to complete an orbit.
If the two planets are alongside one another on a certain day,
then one year later, Earth will be back at the same place, but
Mars will have moved on, and it will take more time for Earth to
finish catching up. Angular velocities add and subtract, just as
velocity vectors do. If the two planets’ angular velocities areω 1
andω 2 , then the angular velocity of one relative to the other is
ω 1 −ω 2. The corresponding period, 1/(1/T 1 − 1 /T 2 ) is known as
the synodic period.
A neutron star example 13
.A neutron star is initially observed to be rotating with an angular
velocity of 2.0 s−^1 , determined via the radio pulses it emits. If its
angular acceleration is a constant−1.0× 10 −^8 s−^2 , how many
rotations will it complete before it stops? (In reality, the angular
acceleration is not always constant; sudden changes often occur,
and are referred to as “starquakes!”)
.The equationvf^2 −vi^2 =2a∆xcan be translated intoω^2 f−ω^2 i=2α∆θ,
giving∆θ= (ω^2 f−ω^2 i)/ 2 α
= 2.0× 108 radians
= 3.2× 107 rotations.4.2.2 Relations between angular quantities and motion of a
point
It is often necessary to be able to relate the angular quantities
to the motion of a particular point on the rotating object. As we
develop these, we will encounter the first example where the advan-
tages of radians over degrees become apparent.
The speed at which a point on the object moves depends on both
the object’s angular velocityωand the point’s distancerfrom the272 Chapter 4 Conservation of Angular Momentum