Chapter 10
Rotational Kinematics and Energy
Dr. Jie Zou PHY 1151G
Department of Physics
1
Outline
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Angular Position
Angular Velocity
Angular Acceleration
Kinematics Equations for Rotations with
Constant Acceleration
Examples
Dr. Jie Zou PHY 1151G
Department of Physics
2
Angular position
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Definition of angular position, :
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: an angle measured from reference
line. The reference line defines  = 0.
Sign convention for angular
position:  > 0 for counterclockwise
rotation from reference line;  < 0 for
clockwise rotation from reference
line.
Units to measure angles: SI units:
radians (rad); other units: degrees (º)
and revolutions (rev).
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1 rev = 360º = 2π rad, 1 rad = 57.3º.
Dr. Jie Zou PHY 1151G
Department of Physics
3
Angular velocity
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Angular displacement  = f - i.
Average angular velocity:
av=/t.
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Sign convention for angular
velocity:
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SI units: radians per second (rad/s).
 > 0 for counterclockwise rotation.
 < 0 for clockwise rotation.
Angular speed: The speed of
rotation or the magnitude of the
angular
Dr. Jie Zou velocity.
PHY 1151G
Department of Physics
4
Period of Rotation
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Definition of period: The time to
complete one revolution, T, is referred to
as the period.
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T = 2 /.
Here  is the angular speed in rad/s.
SI units for T: second (s).
Dr. Jie Zou PHY 1151G
Department of Physics
5
Exercise 10-1 and 10-2
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(a) An old record player rotates
clockwise at 33 rpm (revolutions
per minute). What is its angular
velocity in rad/s?
(b) If a CD rotates at 22.0 rad/s,
what is its angular speed in rpm?
(c) Find the period of a record that
is rotating at 45 rpm.
Dr. Jie Zou PHY 1151G
Department of Physics
6
Angular acceleration
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Average angular acceleration
av =  /t = (f - i)/ t
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SI units: radians per second per
second (rad/s2).
Determination of the sign of the
angular acceleration:
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If  and  have the same sign,
the speed of rotation (angular
speed) is increasing.
If  and  have opposite signs,
the speed of rotation (angular
speed) is decreasing.Dr. Jie Zou PHY 1151G
Department of Physics
7
Exercise 10-3
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As the wind dies, a windmill that was
rotating at 2.1 rad/s begins to slow
down with a constant angular
acceleration of 0.45 rad/s2. How long
does it take for the windmill to come to a
complete stop?
Dr. Jie Zou PHY 1151G
Department of Physics
8
Kinematics Equations for
Rotational Motions
Linear-to-angular analogies
Linear Quantity
x
v
a
Angular Quantity



Linear Equation (a = constant) Angular equation ( = constant)
v = v0 + t
x = x0 + v0 t + at2/2
v2 = v02 + 2a(x - x0)
 =  0 + t
 =  0 +  0 t + t2/2
 2 =  02 + 2( - 0)
Dr. Jie Zou PHY 1151G
Department of Physics
9
Example 10-1
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To throw a curve ball, a
pitcher gives the ball an
initial angular speed of 36.0
rad/s. When the catcher
gloves the ball 0.595 s later,
its angular speed has
decreased (due to air
resistance) to 34.2 rad/s. (a)
What is the ball’s angular
acceleration, assuming it to
be constant? (b) How many
revolutions does the ball
make before being caught?
Dr. Jie Zou PHY 1151G
Department of Physics
10
Active Example 10-1:
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A pulley rotating in the
counterclockwise direction is
attached to a mass suspended
from a string. The mass causes the
pulley’s angular velocity to
decrease with a constant angular
acceleration  = -2.10 rad/s2. (a) If
the pulley’s initial angular velocity
is  0 = 5.40 rad/s, how long does
it take for the pulley to come to
rest? (b) Through what angle does
the pulley turn this time?
Dr. Jie Zou PHY 1151G
Department of Physics
11
Homework
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See online homework assignment on
www.masteringphysics.com
Dr. Jie Zou PHY 1151G
Department of Physics
12