Rotation of a Rigid Body Chapter 10

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Rotation of a Rigid Body (Chapter 10)
Each particle travels in a circle. The speeds of
the particles differ, but each one completes a
full revolution in the same time.
We describe the rotational motion using angle,
angular velocity, and angular acceleration
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Units by convention, angles are measured in
radians.
arc length
s
r
?
r
2p rad 360o
Angular velocity has units of rad/s or
s-1 Angular acceleration has units of rad/s2 or
s-2
(The radian is a ratio of two lengths, and not
really a unit. Some equations will require angles
to be in radians.)
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(radians) (rad/s)
(rad/s2 )
angle (theta) angular velocity
(omega) angular acceleration (alpha)
0
reference axis
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Linear and angular quantities
A particle P travels in a circle of radius r.
The velocity is tangential to the circle and
perpendicular to the radius.
Distance s rq Tangential Velocity
Tangential Acceleration
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Quiz
The Earth rotates on its axis. How does its
angular velocity w vary with location?
a) w is larger at the equator, and smaller near
the poles b) w is smaller at the equator, and
larger near the poles c) w is the same at the
equator and near the poles
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In simpler notation
The tangential component at is equal to the rate
of increase of speed. There is also a radial
(centripetal) component, due to the change in
direction of v
These relations require angular quantities to be
measured in radians (or rad/s, etc.).
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Quiz
Several pennies are placed on a turntable. As the
angular velocity of the turntable is slowly
increased, which penny slides first?
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Quiz
High-speed CD-ROM drives sometimes specify that
they use a constant linear velocity method of
recording. What does this mean for the rotation
rate in revolutions per minute as the write head
moves from the inner tracks to the outer tracks?
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Constant angular acceleration
All for constant a only !
These expressions are should remind you of
relations for constant linear acceleration q
replaces x, w replaces v, a replaces a.
All for constant a only !
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(Extra-derivation) Constant angular acceleration
In this special case, there are relations among
q, w, and a that are analagous to the relations
among x, v, and a for linear motion with constant
linear acceleration.
(constant a initial at t 0)
or
or
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• Example A computer disc has a linear velocity of
  1.3m/s. What is the angular velocity when at the
  innermost track where r23mm.

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Example A computer disc starts from rest and
reaches a final rotation rate of 7200 rev/min
after 10 seconds. Assuming constant angular
acceleration, through how many revolutions does
it turn?
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Summary

• Rotational motion can be described by angle,
  angular velocity, and angular acceleration
• For constant angular acceleration (a special
  case!), kinematical relations are similar to
  those for linear motion with constant linear
  acceleration

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• Practice problems Chapter 10
  Problems 1, 5, 11 (5th ed) Problems 5, 7, 11