Physics 170 - Mechanics
Lecture 22
Rotational Kinematics
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A Particle in
Uniform Circular Motion
For a particle in uniform circular
motion, the velocity vector v remains
constant in magnitude, but it
continuously changes its direction.
However, at each position it is tangent
to the circular path. For this reason, it
is called the tangential velocity of the
particle.
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Angular Position θ
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Angular Position θ
Degrees and revolutions:
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Angular Position θ
Arc length s,
measured in radians:
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Angular Velocity ω
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Rotational Period T
Example: Find the period of a music phonograph record
that is rotating at 45 RPM.
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Angular Acceleration α
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Example: Decelerating Windmill
As the wind dies, a windmill that had
been rotating at ω = 2.1 rad/s begins to
slow down at a constant angular
acceleration of α = −0.45 rad/s2.
How long does it take for the windmill to
come to a complete stop?
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Summary of Angular Variables
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Rotational vs. Linear Kinematics
Analogies between linear and rotational
kinematics:
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Example: Thrown for a Curve
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?
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Connections Between
Linear & Rotational Quantities
The tangential
velocity vt is zero at
the center of rotation
and increases linearly
with r.
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Connections Between
Linear & Rotational Quantities
Question: Two children ride a merrygo-round, with Child 1 at a greater
distance from the axis of rotation
than is Child 2.
How do the angular speeds ω1,2 of
the two children compare?
(a) ω1>ω2
(b) ω1=ω2
(c) ω1<ω2
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Connections Between
Linear & Rotational Quantities
Question: Two children ride a merrygo-round, with Child 1 at a greater
distance from the axis of rotation
than is Child 2.
How do the angular speeds ω1,2 of
the two children compare?
(a) ω1>ω2
(b) ω1=ω2
(c) ω1<ω2
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Connections Between
Linear & Rotational Quantities
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Connections Between
Linear & Rotational Quantities
This merry-go-round
has both tangential and
centripetal acceleration.
Speeding up
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Example: Time to Rest
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 α = −2.10 rad/s2.
(a) If the pulley’s initial angular velocity is ω0 = 5.40
rad/s, how long does it take for the pulley to come to
rest?
(b) Through what angle does the pulley turn during
this time?
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Example: A Rotating Crankshaft
A car’s tachometer indicates the angular velocity
ω of the engine’s crankshaft in rpm. A car stopped at
a traffic light has its engine idling at 500 rpm. When the
light turns green, the crankshaft’s angular velocity speeds up
at a constant rate to 2,500 rpm in a time interval of 3.0 s.
How many revolutions does the crankshaft make in this time
interval?
41.9
5.0 s
41.9
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Example: CD Speed
Unlike old phonograph records that turned with a constant angular speed (like
33 1/3 rpm), CDs and DVDs turn with a variable ω that keeps the tangential
speed vt constant.
Find the angular speed ω that a CD
must have in order to give it a linear
speed vt = 1.25 m/s when the laser
beam shines on the disk
(a) at 2.50 cm from its center, and
(b) at 6.00 cm from its center.
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