Rotational Kinematics and Energy

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Rotational Kinematics and Energy
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• Angular Position, Velocity, and Acceleration
• Rotational Kinematics
• Connections Between Linear and Rotational
  Quantities
• Rolling Motion
• Rotational Kinetic Energy and the Moment of
  Inertia
• Conservation of Energy

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Angular Position, Velocity, and Acceleration
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Angular Position, Velocity, and Acceleration
Degrees and revolutions
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Angular Position, Velocity, and Acceleration
Arc length s, measured in radians
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Angular Position, Velocity, and Acceleration
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Angular Position, Velocity, and Acceleration
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Angular Position, Velocity, and Acceleration
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Angular Position, Velocity, and Acceleration
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Rotational Kinematics
If the angular acceleration is constant
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Rotational Kinematics
Analogies between linear and rotational
kinematics
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Connections Between Linear and Rotational
Quantities
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Connections Between Linear and Rotational
Quantities
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Connections Between Linear and Rotational
Quantities
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Connections Between Linear and Rotational
Quantities
This merry-go-round has both tangential and
centripetal acceleration.
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Rolling Motion
If a round object rolls without slipping, there
is a fixed relationship between the translational
and rotational speeds
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Rolling Motion
We may also consider rolling motion to be a
combination of pure rotational and pure
translational motion
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Rotational Kinetic Energy and the Moment of
Inertia
For this mass,
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Rotational Kinetic Energy and the Moment of
Inertia
We can also write the kinetic energy as
Where I, the moment of inertia, is given by
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Rotational Kinetic Energy and the Moment of
Inertia
Moments of inertia of various regular objects can
be calculated
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Conservation of Energy
The total kinetic energy of a rolling object is
the sum of its linear and rotational kinetic
energies
The second equation makes it clear that the
kinetic energy of a rolling object is a multiple
of the kinetic energy of translation.
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Conservation of Energy
If these two objects, of the same mass and
radius, are released simultaneously, the disk
will reach the bottom first more of its
gravitational potential energy becomes
translational kinetic energy, and less rotational.
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Summary

• Describing rotational motion requires analogs to
  position, velocity, and acceleration

• Average and instantaneous angular velocity

• Average and instantaneous angular acceleration

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Summary

• Period
• Counterclockwise rotations are positive,
  clockwise negative
• Linear and angular quantities

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Summary

• Linear and angular equations of motion

Tangential speed Centripetal acceleration Tangen
tial acceleration
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Summary

• Rolling motion
• Kinetic energy of rotation
• Moment of inertia
• Kinetic energy of an object rolling without
  slipping
• When solving problems involving conservation of
  energy, both the rotational and linear kinetic
  energy must be taken into account.