Chapter 10: Rotational Motional About a Fixed Axis

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Chapter 10 Rotational Motional About a Fixed Axis

• Review of Angular Quantities Motion and Torque
  (10-1,10-2,10-3, 10-5)
• Solving Problems in Rotational Dynamics (10-6,
  10-7)
• Determining Moments of Inertia, Conservation of
  Angular Momentum, and Rotational Kinetic Energy
  (10-8,10-9, 10-10)
• Rotational Translational Motion
  (10-11, 10-12)

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Example 1 The Torque on a Braking Car.

• Interestingly enough when a car brakes the force
  on the front brakes exceed the force on the back
  brakes.
• This is related to the torque around the center
  of mass.
• Consider the car at right which has a mass of
  M1200kg and is decelerating at 0.50g.

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The Line Integral for Rotational Work

• The work done on a rotating body about a fixed
  axis can written in terms of angular quantities.
• Suppose a force F is exerted at a distance from
  the axis of rotation as shown in the figure.

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Rotational Translational Motion

• An object in translation has kinetic energy which
  can be described by the total mass M and the
  velocity of the CM vCM
• KE (1/2)M vCM2
• An object rotating about an axis through its
  center of mass has rotational kinetic energy
  given by
• KE (1/2)ICM w2
• An object such as a rolling wheel has both!
• KE (1/2)M vCM2(1/2)ICM w2

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Example 2 Rolling Sphere

• A solid sphere of mass M and radius R starts at
  rest and rolls down an incline of height H.
• Ignoring friction and assuming there is no
  slippage whats the final velocity?

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Example 3 What about a Hoop or a Cylinder?
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Proof of KE(1/2)M vCM2(1/2)ICM w2

• The most general statement is the total kinetic
  energy of a moving body will be equal to the
  translational kinetic energy of its CM plus the
  kinetic energy of the motion of the object
  relative to the center of mass.
• Starting with the figure at the right, this is
  really a vector argument.

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Example 4 A Yo-yos Acceleration!

• Approximate the shape of a yo-yo as that of a
  solid cylinder of mass M and radius R.
• As it falls and unwinds the string find the
  acceleration and tension in the string.

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Example 5 How Long Does It Take for a Slipping
Ball to Roll?

• A bowling ball of mass and radius R is thrown
  down the alley. Initially it slides with linear
  speed vo. As it slides it begins to roll.
• How long before it is completely rolling with out
  slipping?

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We Are in the Stretch!

• There are four lessons left involving Chapter 11
  and a bit of Chapter 12.
• In Chapter 11 we continue to explore angular
  momentum
• In Chapter 12 we study equilibrium.
• The next Quiz is May 2. You know the drill by
  now!