Physics 2211: Lecture 36 Todays Agenda

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Physics 2211 Lecture 36Todays Agenda

• Kinetic energy of a rotating system
• Moment of inertia
• Discrete particles
• Continuous solid objects
• Parallel axis theorem

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A 3 kg object moving in the x direction has a
one-dimensional elastic collision with a 5.0 kg
object initially at rest. After the collision,
the 5.0 kg object has a velocity of 6.0 m/s in
the x direction. What was the initial speed of
the 3.0 kg object? (1) 6.0 m/s (2) 4.5 m/s
(3) 7.0 m/s (4) 3.5 m/s (5) 9.5 m/s (6) 8.0
m/s
17308
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A 0.4 kg softball has a velocity of 15 m/s, at an
angle of q50 degrees as shown, just before
making contact with a bat. The ball leaves the
bat 0.2 s later moving horizontally with a speed
of 20 m/s. What is the magnitude of the average
resultant force on the ball while in contact
with the bat? (1) 320 N (2) 82 N (3) 31
N (4) 77 N (5) 65 N (6) 59 N
50 degrees
17163
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A 12 m uniform board of mass 50 kg rests on a
horizontal frictionless surface. Both the board
and a person standing on one end are initially at
rest. When the person walks to the opposite end
of the plank and stops, the plank moves 7 m
relative to the surface. What is the mass of
the person? (1) 70 kg (2) 86 kg (3) 62
kg (4) 93 kg (5) 55 kg (6) 78 kg
17515
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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

• 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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Moment 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

(1) Ia gt Ib gt Ic (2) Ia gt Ic gt Ib (3)
Ib gt Ia gt Ic
a
b
c
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Moment of Inertia

• Label masses and lengths

m
a
L
b
So (2) 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

• 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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Moment 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?

(1) solid aluminum (2) hollow gold (3) same
hollow
solid
same mass radius
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Moment 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...

• Some examples of I for solid objects

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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Recap of todays lecture

• Kinetic energy of a rotating system
• Moment of inertia
• Discrete particles
• Continuous solid objects