Rotational Dynamics

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Rotational Dynamics

• Speed of a rolling object at the bottom of an
  inclined plane

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Rotational Dynamics

• Speed of a rolling object at the bottom of an
  inclined plane
• Recall
• Using conservation of energy we were able to find
  the speed of an object dropped from a vertical
  height, h, just before it hits bottom
• Total mechanical energy at the top mgh
• Total mechanical energy at the bottom

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Rotational Dynamics
Total mechanical at top mgh
Ignoring friction
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Rotational Dynamics
mass cancels
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Rotational Dynamics
This also works with a translating object down an
inclined plane
(frictionless)
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Rotational Dynamics
Now, take a rolling object say a solid sphere
- (bowling ball)
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Rotational Dynamics

• Whether it rolls down the incline
• which includes translation
• Or just translates
• Objects at the top of the incline have the same
  mechanical energy
• PE mgh

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Rotational Dynamics

• But for a rolling object
• Some of that energy at the top is going into
  rolling the object
• With the remainder translating the object.

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Rotational Dynamics

• Now, solve for translational speed at bottom, v

• From Table 8.3, p. 254
• Mass moment of inertia of a solid sphere

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Rotational Dynamics

• Simplify

• Now, since we want v, lets work on getting rid of
  the angular speed, ?

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Rotational Dynamics

• For a rolling, non slipping object, the
  tangential speed of a point on the outer radius
• Is equal to the translational speed of the center
  of mass --- axis of rotation
• And tangential speed is related to rotational
  speed by

• For one revolution, the point on the radius
  traveled one circumference
• Equal to the distance the center of mass traveled

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• Which reduces to

• Notice the radius cancels!!

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• AND, masses cancel, leaving

• So, mass and radius do not make a difference!!

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Rotational Dynamics

• Which simplifies to

• Solving for v

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Rotational Dynamics

• For other rotational objects such as hoops, solid
  cylinders, etc.
• I leave it up to you to derive the velocity
• Start with several slides ago

• Insert the appropriate mass moment of inertia
  from Table 8.3, p. 254

• And solve for v

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