Rotational Motion

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Rotational Motion
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WORKING WITH ROTATIONAL KINEMATICS
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Rotational Motion

• Translational motion is the motion of an object
  through space.
• Rotational motion is the spinning of an object
  around an axis.
• Objects may have purely translational motion,
  purely rotational motion, or both.

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Rotational Motion

• A rigid body is composed of particles in fixed
  positions.
• In a rigid body undergoing purely rotational
  motion, all points of the body move in circles
  centered around a line called the axis of
  rotation.

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Rotational Motion

• Key to learning rotational motion
• THE CONCEPTS FROM TRANSLATIONAL MECHANICS (E.G.
  VELOCITY, ACCELERATION, FORCE ENERGY, MOMENTUM)
  HAVE ANALOGOUS ROTATIONAL CONCEPTS.

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Rotational Kinematics
Angular displacement (?) is the rotational analog
of displacement (?x).
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Rotational Kinematics

• Angular displacements are measured in radians.
• One radian (or rad) is defined as the angle
  subtended by an arc length equal to one radius.
  (See next slide)
• Subtended
• A curve whose endpoints lie on the rays of an
  angle. Equal arc lengths subtend equal angles

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Rotational Kinematics

• ? Arc length/radius
• ? r/r 1 radian

?
For one complete revolution ? 2?r/r 2?
radians So, 2? rad 360
? 1 rad
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Rotational Kinematics

• Answers
• 180
• ?/2
• 57.3
• 20?

• Find the following
• ? rad ___ degrees
• 90 ___ rad
• 1 rad ___ degrees
• 10 rev ___ rad

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Rotational Kinematics

• Average angular velocity (?avg) is the rotational
  analog of velocity (v).
• Velocity Displacement / Time
• v ?x / t
• Ang. velocity Ang. displacement / Time
• ?avg ? / t

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Rotational Kinematics

• The angular velocity is the same for all points
  on a rigid body (Why?)
• Each point on a rigid point moves through the
  same angle (?)
• What are the units for angular velocity?
• Degrees/time radians/s
• Why does angular displacement divided by time
  calculate AVERAGE angular velocity?
• (Change in angular position)/time ?avg

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Rotational Kinematics

• What is the angular velocity of the points on a
  record rotating at 45 rpm?
• ? 45 rev x (2? rad / 1 rev) 90? rad
• ? ? / t
• ? 90? rad / 60 s
• ? 3?/2 rad/s

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Rotational Kinematics

• Angular acceleration (?) is the rotational analog
  of acceleration (a).
• Acceleration Change in velocity / Time
• a ?v / t
• Ang. accel. Change in Ang.veloc. / Time
• ? ?? / t

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Rotational Kinematics

• The angular acceleration is the same for all
  points on a rigid body (Why?)
• What are the units for angular acceleration?
• Technically, change in angular velocity divided
  by time calculates AVERAGE angular acceleration.
  However, we only study constant angular
  accelerations.

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Rotational Kinematics

• A record begins spinning from rest and reaches a
  rotation of 45 rpm in three seconds. What is the
  angular acceleration of the points on the record?
• ? ?? / t
• ? (3?/2 rad/s - 0) / 3 s
• ? ?/2 rad/s2

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Rotational Kinematics

• In translational motion, we used several
  kinematics equations to analyze situations of
  constant acceleration.
• In rotational motion, we have equivalent
  kinematics equations to analyze situations of
  constant angular acceleration.

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Rotational Kinematics

• Rotational Motion
• ? ?o ?t
• ?? (1/2)(?o ?)t
• ?avg (1/2)(?o ?)
• ? ?ot (1/2)?t2
• ?2 ?o2 2??

• Linear Motion
• v vo at
• ?x (1/2)(vo v)t
• vavg (1/2)(vo v)
• ?x vot (1/2)at2
• v2 vo2 2a?x

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• Rotational Motion Terms
• ?z Refers to rotation about the z-axis
• This can be positive or negative (See p333)
• ?z Acceleration along z axis. Can be pos or neg
• Use Right Hand Rule to determine direction (z or
  z)
• See p333

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Rotation with (non-constant) Angular
Acceleration
This graph shows the rotational ?z and az vs t
for a particular rotating body Q1 During which
time interval is the object speeding up? Q2
During which time intervals is it slowing down?
0-2, 4-6
2-4
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Problem 1
Given ? 27.5 rad/s at t 0 a -10
rad/s2 Line PG lies on the disc
along the x -axis

• Find
• ? at t 0.30 s?
• Angle PQ makes with x axis at this time?

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Problem 1
In addition to stated givens we know Ti 0 at t
0s
Given ? 27.5 rad/s at t 0 a -10
rad/s2 Line PG lies on the disc
along the x -axis
Part a Solution Use ?z ?iz azt
for t 0.3 s ?z (27.5 rad/s)
(-10 rad/s2)(0.3s) 24.5 rad/s
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Part b
In addition to stated givens we know Ti 0 at t
0s
Part b Solution Use T Ti ?izt ½ azt2
for t 0.3 s T 0 (27.5
rad/s)(0.3s) ½ (-10 rad/s2)(0.3s)2 7.8 rad
revolutions? 7.8 rad (1rev/2p rad) 1.24 rev
The disc has gone through one revolution plus
0.24 into the next. So the angle at that time
with the x axis is 0.24 (360/rev) 87
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Homework Practice
p353 Chapter 9 Do 1,3,7,8,12,14,19 READ 9.1-9.4