Rotation

1
Chapter 10 Rotation
2

• Rotation of a rigid body
• We consider rotational motion of a rigid body
  about a fixed axis
• Rigid body rotates with all its parts locked
  together and without any change in its shape
• Fixed axis it does not move during the rotation
• This axis is called axis of rotation
• Reference line is introduced

3

• Angular position
• Reference line is fixed in the body, is
  perpendicular to the rotation axis, intersects
  the rotation axis, and rotates with the body
• Angular position the angle (in radians or
  degrees) of the reference line relative to a
  fixed direction (zero angular position)

4

• Angular displacement
• Angular displacement the change in angular
  position.
• Angular displacement is considered positive in
  the CCW direction and holds for the rigid body as
  a whole and every part within that body

5

• Angular velocity
• Average angular velocity
• Instantaneous angular velocity the rate of
  change in angular position

6

• Angular acceleration
• Average angular acceleration
• Instantaneous angular acceleration the rate of
  change in angular velocity

7

• Rotation with constant angular acceleration
• Similarly to Chapter 2 (case of 1D motion with a
  constant acceleration) we can derive a set of
  formulas (Table 10-1)

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Chapter 10 Problem 6
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• Relating the linear and angular variables
  position
• For a point on a reference line at a distance r
  from the rotation axis
• ? is measured in radians

10

• Relating the linear and angular variables speed
• ? is measured in rad/s
• Period (recall Ch. 4)

11

• Relating the linear and angular variables
  acceleration
• a is measured in rad/s2
• Centripetal acceleration (Ch. 4)

12

• Rotational kinetic energy
• We consider a system of particles participating
  in rotational motion
• Kinetic energy of this system is
• Then

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• Moment of inertia
• From the previous slide
• Defining moment of inertia (rotational inertia)
  as
• We obtain for rotational kinetic energy

14

• Moment of inertia rigid body
• For a rigid body with volume V and density ?(V)
  we generalize the definition of a rotational
  inertia
• This integral can be calculated for different
  shapes and density distributions
• For a constant density and the rotation axis
  going through the center of mass the rotational
  inertia for 9 common body shapes is given in
  Table 10-2 (next slide)

15
Moment of inertia rigid body
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• Moment of inertia rigid body
• The rotational inertia of a rigid body depends
  on the position and orientation of the axis of
  rotation relative to the body
• More information at http//scienceworld.wolfram.
  com/physics/MomentofInertia.html

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• Parallel-axis theorem
• Rotational inertia of a rigid body with the
  rotation axis, which is perpendicular to the xy
  plane and going through point P
• Let us choose a reference frame, in which the
  center of mass coincides with the origin

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Parallel-axis theorem
19
Parallel-axis theorem
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Chapter 10 Problem 39
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• Torque
• We apply a force at point P to a rigid body that
  is free to rotate about an axis passing through O
• Only the tangential component Ft F sin f of
  the force will be able to cause rotation

22

• Torque
• The ability to rotate will also depend on how
  far from the rotation axis the force is applied
• Torque (turning action of a force)
• SI unit Nm (dont confuse with J)

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• Torque
• Torque
• Moment arm r- r sinf
• Torque can be redefined as
• force times moment arm
• t F r-

24

• Newtons Second Law for rotation
• Consider a particle rotating under the influence
  of a force
• For tangential components
• Similar derivation for rigid body

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Chapter 10 Problem 51
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• Rotational work
• Work
• Power
• Work kinetic energy theorem

27
Corresponding relations for translational and
rotational motion (Table 10-3)
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Answers to the even-numbered problems Chapter
10 Problem 2 14 rev
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• Answers to the even-numbered problems
• Chapter 10
• Problem 10
• 30 s
• (b) 1.8 103 rad

30

• Answers to the even-numbered problems
• Chapter 10
• Problem 22
• 3.0 rad/s
• (b) 30 m/s
• (c) 6.0 m/s2
• (d) 90 m/s2

31

• Answers to the even-numbered problems
• Chapter 10
• Problem 36
• 7.1
• (b) 64

32

• Answers to the even-numbered problems
• Chapter 10
• Problem 46
• 8.4 N m
• (b) 17 N m
• (c) 0

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Answers to the even-numbered problems Chapter
10 Problem 50 1.28 kg m2
34

• Answers to the even-numbered problems
• Chapter 10
• Problem 64
• 0.15 kg m2
• (b) 11 rad/s

35

• Answers to the even-numbered problems
• Chapter 10
• Problem 78
• 1.57m/s2
• 4.55 N
• (c) 4.94 N