Rotational Kinematics and Inertia

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Rotational Kinematics and Inertia
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Circular Motion

• Angular displacement Dq q2-q1
• How far it has rotated
• Units radians 2p 1 revolution
• Angular velocity w Dq/Dt
• How fast it is rotating
• Units radians/second 2p 1 revolution
• Angular acceleration is the change in angular
  velocity w divided by the change in time. a
  Dw/Dt
• How much is it speeding up or slowing down
• Units radians/second2

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Period, Frequency

• Frequency
• Number of revolutions per sec
• Period 1/frequency T 1/f 2p / w
• Time to complete 1 revolution

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Circular to Linear(Why use Radians)

• Displacement Ds r Dq
• (q in radians)
• Speed v Ds/Dt r Dq/Dt rw
• Direction of v is tangent to circle
• Acceleration a ra

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Angular Acceleration

• If the speed of a roller coaster car is 15 m/s at
  the top of a 20 m loop, and 25 m/s at the bottom.
  What is the cars average angular acceleration if
  it takes 1.6 seconds to go from the top to the
  bottom?

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Comparison to 1-D kinematics

• Angular Linear

And for a point at a distance R from the rotation
axis
x R????????????v ?R ??????????a ?R
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Example cd player

• The CD in your disk player spins at about 20
  radians/second. If it accelerates uniformly from
  rest with angular acceleration of 15 rad/s2, how
  many revolutions does the disk make before it is
  at the proper speed?

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Example 48x cd-rom

• A 48x cd-rom spins at about 9600 rpm. If it
  takes 1.5 sec. to get up to speed, what is the
  angular acceleration? How many revolutions does
  the disk make before it is at the proper speed?

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Rotational Inertia, I

• Tells how difficult it is get object spinning.
  Just like mass tells you how difficult it is to
  get object moving.
• Fnet m a Linear Motion
• tnet I a Rotational Motion
• I S miri2 (units kg m2)
• Note! Rotational Inertia depends on what you are
  spinning about (basically the ri in the equation).

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Inertia Rods

• Two batons have equal mass and length.
• Which will be easier to spin
• A) Mass on ends
• B) Same
• C) Mass in center

I S m r2 Further mass is from axis of
rotation, greater moment of inertia (harder to
spin)
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Example baseball bat
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Rotational Inertia Table

• For objects with finite number of masses, use I
  S m r2. For continuous objects, use table below.

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