Rock and Roll

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Rock and Roll
Lecture 44 4/25/05
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Translational rotational motion combined

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

VCM
?
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Rolling Motion

• Rolling without slipping

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When an object rolls without slipping the point
of contact with the ground does not move relative
to the ground. This means that any force of
friction exerted on the object by the surface is
a static friction force.
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Consider a disk that rolls down an inclined
surface. What is the translation acceleration of
the disk?
The force of friction does no work as there is
no motion of the disk at the point of contact.
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(No Transcript)
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ExampleRotations

• Two uniform cylinders are machined out of solid
  aluminum. One has twice the radius of the other.
• If both are placed at the top of the same ramp
  and released, which is moving faster at the
  bottom?

(1) bigger one (2) smaller one (3) same

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Example Solution
Answer (3) same speed does not depend on size,
as long as the shape is the same!!
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A rolling object has two forms of kinetic energy.
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What is the total kinetic energy of sphere
(radius 5.0 cm, mass750 g ) rolling without
slipping with a speed of 30 m/s?
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A hoop and disk roll without slipping down a
10-meter long ramp that inclined 35 degrees with
the horizontal. What is the speed of each object
at the bottom of the ramp? Assume each object
has a mass m and radius r and they start from
rest.
Use conservation of energy and set the bottom of
the ramp as y0.
Hoop
Disk
Note result is not a function of mass or radius!
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A sphere (mass M, radius R) pivots about a point
on its surface. What is the rotational velocity
of the sphere after it rotates 90 degrees?
Use parallel axis theorem to compute I.
x
The center of mass falls a distance R
x
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The center of mass of the bar falls a distance
L/2. This decrease in potential energy results
in an increase of rotational kinetic energy.
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• A solid sphere (radius 10 cm, mass5.0 kg) is
  attached to the end of a rod (length 50 cm, mass
  2.0 kg). For the object thus formed what are
• The center of mass
• The moment of inertia

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The center of mass of the rod is at its midpoint.
The center of mass of the sphere is at its
center. Measured from the left end the center of
the rod is 25 cm to the right and the center of
the sphere is 60 cm to the right.
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Use parallel axis theorem to compute the moment
of inertia about the pin in the left end.
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The composite object is released from rest and
swings from 20 degrees below the horizontal to a
vertical position. What is it angular speed when
it reaches the vertical position?
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The pendulum is hung vertically and is free to
rotate about a pin in its upper end. What is the
period for small oscillations about this position?
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d is the distance from the pin to the center of
mass.
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