Physics 106P: Lecture 21 Notes

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Which way is she turning (viewed from top)
A) Clockwise B) Counter-clockwise
Hi Prof Selen I know this does not have much to
do with physics per se (well, maybe, the person
is rotating with some ... velocity) but I thought
it was interesting, maybe to use the i-clicker to
see which way most people see or
whatnot... Anyway, the link is http//www.news.co
m.au/dailytelegraph/story/0,22049,22535838-5012895
,00.html Regards Jurand
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Physics 211 Lecture 17Todays Agenda

• Rotational Kinematics
• Analogy with one-dimensional kinematics
• Kinetic energy of a rotating system
• Moment of inertia
• Discrete particles
• Continuous solid objects
• Parallel axis theorem

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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 pulleys are without mass.
• Rotation is extremely important, however, and we
  need to understand it!
• Most of the equations we will develop are simply
  rotational analogues of ones we have already
  learned when studying linear kinematics and
  dynamics.

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Lecture 17, Act 1Rotations

• Bonnie sits on the outer rim of a merry-go-round,
  and Klyde sits midway between the center and the
  rim. The merry-go-round makes one complete
  revolution every two seconds.
• Klydes angular velocity is

(a) the same as Bonnies (b) twice
Bonnies (c) half Bonnies
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Lecture 17, Act 1Rotations

• The angular velocity w of any point on a solid
  object rotating about a fixed axis is the same.
• Both Bonnie Klyde go around once (2p radians)
  every two seconds.

(Their linear speed v will be different since v
wr).
w
6
Rotational Variables.
Spin round blackboard

• 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...

• 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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Rotational Variables...
constant
v
x
R
?
?

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

?
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Summary (with comparison to 1-D kinematics)

• 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

• A wheel with radius R 0.4 m 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 4 m/s2. How many revolutions
  has the wheel made after 10 seconds?
  (One revolution 2? radians)

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Wheel And Rope...

• Use a ?R to find ?
• ?? a / R 4 m/s2 / 0.4 m 10 rad/s2

• Now use the equations we derived above just as
  you would use the kinematic equations from the
  beginning of the semester.

a
?
R
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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

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Rotation Kinetic Energy...

• So but vi ?ri

I has units of kg m2.
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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
Inertia Rods

• 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

• 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
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Calculating Moment of Inertia...

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

Using the Pythagorean Theorem
so
L/2
m
m
r
L
m
m
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Calculating Moment of Inertia...

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

r
L
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Calculating Moment of Inertia...

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

r
m
m
L
m
m
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Calculating Moment of Inertia...

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

I 2mL2
I mL2
I 2mL2
m
m
L
m
m
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Lecture 17, 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 17, 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...

• 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
Hoop

• 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.
R
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Moments of Inertia...
Sphere and disk

• Some examples of I for solid objects

Solid sphere of mass M and radius R, about an
axis through its center.
R
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Lecture 17, Act 3Moment of Inertia

• Two spheres have the same radius and equal
  masses. One is made of solid aluminum, and the
  other is made from a hollow shell of gold.
• Which one has the biggest moment of inertia about
  an axis through its center?

(a) solid aluminum (b) hollow gold (c) same
hollow
solid
same mass radius
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Lecture 17, Act 3Moment of Inertia

• Moment of inertia depends on mass (same for both)
  and distance from axis squared, which is bigger
  for the shell since its mass is located farther
  from the center.
• The spherical shell (gold) will have a bigger
  moment of inertia.

ISOLID lt ISHELL
hollow
solid
same mass radius
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Moments of Inertia...
Rod

• Some examples of I for solid objects (see also
  Tipler, Table 9-1)

Thin rod of mass M and length L, about a
perpendicular axis through its center.
L
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Parallel Axis Theorem

• Suppose the moment of inertia of a solid object
  of mass M about an axis through the center of
  mass, ICM, is known.
• The moment of inertia about an axis parallel to
  this axis but a distance D away is given by
• IPARALLEL ICM MD2
• So if we know ICM , it is easy to calculate the
  moment of inertia about a parallel axis.

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Parallel Axis Theorem Example

• Consider a thin uniform rod of mass M and length
  D. Figure out the moment of inertia about an axis
  through the end of the rod.
• IPARALLEL ICM MD2

DL/2
M
CM
x
L
ICM
IEND
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Connection with CM motion

• Recall what we found out about the kinetic energy
  of a system of particles in Lecture 15

KREL
KCM

• For a solid object rotating about its center of
  mass, we now see that the first term becomes

Substituting
but
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Connection with CM motion...

• So for a solid object which rotates about its
  center or mass and whose CM is moving

VCM
?
We will use this formula more in coming lectures.
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Recap of todays lecture

• Rotational Kinematics
• Analogy with one-dimensional kinematics
• Kinetic energy of a rotating system
• Moment of inertia
• Discrete particles
• Continuous solid objects
• Parallel axis theorem