v = r?

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v r? atan ra arad r?2
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Rotation about a Moving Axis

• Every motion of of a rigid body can be
  represented
• as a combination of motion of the center of mass
  (translation) and rotation about an axis through
  the center of mass
• The total kinetic energy can always be
  represented as the sum of a part associated with
  motion of the center of mass (treated as a point)
  plus a part asociated with rotation about an axis
  through the center of mass

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Total Kinetic Energy

• Ktotal (1/2)Mvcm2 (1/2)Icm?2

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CHAPTER 10 DYNAMICS OF ROTATIONAL MOTION
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A plumbing problem
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MID-TERM II MOVED TO NOVEMBER 14th
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Help Session
November 12th
MAP 306 5-7 PM
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Torque on a pulley
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Unwinding a winch
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Rotation about a moving axis
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Conservation of angular momentum

• When the sum of the torques of all external
• forces acting on a system is zero, then
• THE TOTAL ANGULAR MOMENTUMIS CONSTANT
  (CONSERVED)

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The professor as figure skater?

• It seems that danger to the instructor is
  proportional to interest in any given
  demonstration.

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Internal forces cannot change the total momentum
of a system.
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Balancing on a teeter-totter Figure 10.25

• The heavier child must sit closer to balance the
  torque from the smaller child.
• Refer to example 10.11 on page 314.

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Balanced forces during exercise
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A gyroscope in the laboratory
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A 1.31-kg bowling trophy is held at arms length,
a distance of 0.505 m from the shoulder joint.
What torque does the trophy exert about the
shoulder if the arm is (a) horizontal, or (b) at
an angle of 20 below the horizontal?
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• A school yard teeter-totter with a total
  length of 5.2 m and a mass of 36 kg is pivoted at
  its center. A 18 kg child sits on one end of the
  teeter-totter.
• Where should a parent push vertically downward
  with a force of 210 N in order to hold the
  teeter-totter level?
• Where should the parent push with a force of 310
  N?
• How would your answer to parts (a) and (b) change
  if the mass of teeter-totter were doubled?

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Ch 9 Problem 39 The flywheel of a gasoline
engine is required to give up 500 J of kinetic
energy while its angular velocity decreases from
650 rev/min to 520 rev/min. What moment of
inertia is required?
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• Ch 9 Problem 49
• A size-5 soccer ball of diameter 22.6 cm and
  mass 426 g rolls up a hill without slipping,
  reaching a maximum height of 5.00 m above the
  base of the hill. We can model this ball as a
  thin-walled hollow sphere.
• At what rate was it rotating at the base of the
  hill?
• (b) How much rotational kinetic energy did it
  then have?

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• Ch 9. Problem 51
• A solid uniform cylinder and a solid uniform
  sphere, each with the same mass and diameter,
  approach a hill rolling with a forward speed of
  6.50 m/s. Both of them roll up the hill without
  slipping.
• Find the maximum height that each of the centers
  will
• reach.
• (b) Why do they reach different heights? Didnt
  both of
• them have the same speed at the bottom of
  the hill?
• (c) Would your answer to part (a) be different
  if these
• two objects (1) did not have the same masses
  or (2)
• did not have the same diameter? What makes
  you
• say this?

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• Ch 10 Problem 27
• A certain drawbridge can be modeled as a
  uniform 15,000 N bar, 12.0 m long, pivoted about
  its lower end. When this bridge is raised to an
  angle of 60.0 degrees above the horizontal, the
  cable holding it suddenly breaks, allowing the
  bridge to fall. At the instant after the cable
  breaks,
• what is this torque on the bridge about the
  pivot
• and
• (b) at what rate is its angular momentum changing?