Physics 151: Lecture 20 Todays Agenda

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Physics 151 Lecture 20Todays Agenda

• Topics (Chapter 10)
• Rotational Kinematics Ch. 10.1-3
• Rotational Energy Ch. 10.4
• Moments of Inertia Ch. 10.5

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Rotation

• Up until now we have gracefully avoided dealing
  with the rotation of objects.
• We have studied objects that slide, not roll.
• We have assumed wheels are massless.
• Rotation is extremely important, however, and we
  need to understand it !
• Most of the equations we will develop are simply
  rotational versions of ones we have already
  learned when studying linear kinematics and
  dynamics.

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RecallKinematic of Circular Motion
y
x
For uniform circular motion
? is angular velocity
Animation
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Example
See text 10.1

• The angular speed of the minute hand of a clock,
  in rad/s, is
• a. p/1800
• b. p/60
• c. p /30
• d. p
• e. 120 p

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Rotational Variables
See text 10.1

• Rotation about a fixed axis
• Consider a disk rotating aboutan axis through
  its center
• First, recall what we learned aboutUniform
  Circular Motion
• (Analogous to )

?
?
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Rotational Variables...
See text 10.1

• Now suppose ? can change as a function of time
• We define the angular acceleration

?

• Consider the case when ?is constant.
• We can integrate this to find ? and ? as a
  function of time

?
?
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Example
See text 10.1

• The graphs below show angular velocity as a
  function of time. In which one is the magnitude
  of the angular acceleration constantly decreasing
  ?

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Rotational Variables...
See text 10.2
constant
v
x
R
?
?

• Recall also that for a point a distanceR away
  from the axis of rotation
• x ?R
• v ?R
• And taking the derivative of this we find
• a ?R

?
Animation
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Summary (with comparison to 1-D kinematics)
See text 10.3

• Angular Linear

And for a point at a distance R from the rotation
axis
x R????????????v ?R ??????????a ?R
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Example Wheel And Rope
See text 10.1

• A wheel with radius R 0.4m rotates freely about
  a fixed axle. There is a rope wound around the
  wheel. Starting from rest at t 0, the rope is
  pulled such that it has a constant acceleration a
  4m/s2. How many revolutions has the wheel made
  after 10 seconds? (One revolution
  2? radians)

a
R
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Example
See text 10.1

• The turntable of a record player has an angular
  velocity of 8.0 rad/s when it is turned off. The
  turntable comes to rest 2.5 s after being turned
  off. Through how many radians does the turntable
  rotate after being turned off ? Assume constant
  angular acceleration.
• a. 12 rad
• b. 8.0 rad
• c. 10 rad
• d. 16 rad
• e. 6.8 rad

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Rotation Kinetic Energy

• Consider the simple rotating system shown below.
  (Assume the masses are attached to the rotation
  axis by massless rigid rods).
• The kinetic energy of this system will be the sum
  of the kinetic energy of each piece

Recall text 9.6, systems of particles, CM
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Rotation Kinetic Energy...

• So but vi ?ri

v1
m4
m1
v4
r1
?
r4
v2
m3
r2
r3
m2
v3
I has units of kg m2.
Recall text 9.6, systems of particles, CM
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Lecture 20, Act 1Rotational Kinetic Energy

• I have two basketballs. BB1 is attached to a
  0.1m long rope. I spin around with it at a rate
  of 2 revolutions per second. BB2 is on a 0.2m
  long rope. I then spin around with it at a rate
  of 2 revolutions per second. What is the ratio of
  the kinetic energy of BB2 to that of BB1?
• A) 1/4 B) 1/2 C) 1 D) 2 E) 4

BB1
BB2
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Rotation Kinetic Energy...

• The kinetic energy of a rotating system looks
  similar to that of a point particle Point
  Particle Rotating System

v is linear velocity m is the mass.
? is angular velocity I is the moment of
inertia about the rotation axis.
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Moment of Inertia
See text 10.4

• So where

• Notice that the moment of inertia I depends on
  the distribution of mass in the system.
• The further the mass is from the rotation axis,
  the bigger the moment of inertia.
• For a given object, the moment of inertia will
  depend on where we choose the rotation axis
  (unlike the center of mass).
• We will see that in rotational dynamics, the
  moment of inertia I appears in the same way that
  mass m does when we study linear dynamics !

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Calculating Moment of Inertia
See text 10.5

• We have shown that for N discrete point masses
  distributed about a fixed axis, the moment of
  inertia is

where r is the distance from the mass to the
axis of rotation.
Example Calculate the moment of inertia of four
point masses (m) on the corners of a square whose
sides have length L, about a perpendicular axis
through the center of the square
m
m
L
m
m
See example 10.4 (similar)
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Calculating Moment of Inertia...
See text 10.5

• The squared distance from each point mass to the
  axis is

L/2
m
m
r
L
m
m
See example 10.4 (similar)
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Calculating Moment of Inertia...
See text 10.5

• Now calculate I for the same object about an axis
  through the center, parallel to the plane (as
  shown)

r
L
See example 10.4 (similar)
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Calculating Moment of Inertia...
See text 10.5

• Finally, calculate I for the same object about an
  axis along one side (as shown)

r
m
m
L
m
m
See example 10.4 (similar)
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Calculating Moment of Inertia...
See text 10.5

• For a single object, I clearly depends on the
  rotation axis !!

I 2mL2
I mL2
I 2mL2
m
m
L
m
m
See example 10.4 (similar)
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Lecture 20, Act 2Moment of Inertia

• A triangular shape is made from identical balls
  and identical rigid, massless rods as shown. The
  moment of inertia about the a, b, and c axes is
  Ia, Ib, and Ic respectively.
• Which of the following is correct

a
(a) Ia gt Ib gt Ic (b) Ia gt Ic gt Ib (c)
Ib gt Ia gt Ic
b
c
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Lecture 20, Act 2Moment of Inertia

• Label masses and lengths

m
a
L
b
So (b) is correct Ia gt Ic gt Ib
L
c
m
m
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Calculating Moment of Inertia...
See text 8-5

• For a discrete collection of point masses we
  found
• For a continuous solid object we have to add up
  the mr2 contribution for every infinitesimal mass
  element dm.
• We have to do anintegral to find I

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Moments of Inertia
See text 10.5

• Some examples of I for solid objects

Thin hoop (or cylinder) of mass M and radius
R, about an axis through its center,
perpendicular to the plane of the hoop.
Thin hoop of mass M and radius R, about an axis
through a diameter.
see Example 10.5 in the text
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Moments of Inertia

• Some examples of I for solid objects

• Solid disk or cylinder of mass M and radius
  R, about a perpendicular axis through its center.

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Moments of Inertia...
See text 10.5

• Some examples of I for solid objects

Solid sphere of mass M and radius R, about an
axis through its center.
R
Thin spherical shell of mass M and radius R,
about an axis through its center.
R
See Table 10.2, Moments of Inertia
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Recap of todays lecture

• Chapter 9,
• Center of Mass
• Elastic Collisions
• Impulse