\Rotational Motion

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\Rotational Motion
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Rotational Inertia and Newtons Second Law

• In linear motion, net force and mass determine
  the acceleration of an object.
• For rotational motion, torque determines the
  rotational acceleration.
• The rotational counterpart to mass is rotational
  inertia or moment of inertia.
• Just as mass represents the resistance to a
  change in linear motion, rotational inertia is
  the resistance of an object to change in its
  rotational motion.
• Rotational inertia is related to the mass of the
  object.
• It also depends on how the mass is distributed
  about the axis of rotation.

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Rotational Inertia and Newtons Second Law

• The resistance to a change in rotational motion
  depends on
• the mass of the object
• the square of the distance of the mass from the
  axis of rotation.
• For an object with its mass concentrated at a
  point
• Rotational inertia mass x square of distance
  from axis
• I mr2
• The total rotational inertia of an object like a
  merry-go-round can be found by adding the
  contributions of all the different parts of the
  object.

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Rotational Inertia and Newtons Second Law

• Newtons second law for linear motion
• Fnet ma
• Newtons second law for rotational motion
• The net torque acting on an object about a given
  axis is equal to the rotational inertia of the
  object about that axis times the rotational
  acceleration of the object.
• ?net I?
• The rotational acceleration produced is equal to
  the torque divided by the rotational inertia.

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Example a baton with a mass at both ends

• Most of the rotational inertia comes from the
  masses at the ends.
• A torque can be applied at the center of the rod,
  producing a rotational acceleration and starting
  the baton to rotate.
• If the masses were moved toward the center, the
  rotational inertia would decrease and the baton
  would be easier to rotate.

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Conservation of Angular Momentum

• How do spinning skaters or divers change their
  rotational velocities?

I mr2
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Angular Momentum

• Linear momentum is mass (inertia) times linear
  velocity p mv
• Angular momentum is rotational inertia times
  rotational velocity
• L I?
• Angular momentum may also be called rotational
  momentum.
• A bowling ball spinning slowly might have the
  same angular momentum as a baseball spinning much
  more rapidly, because of the larger rotational
  inertia I of the bowling ball.

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Conservation of Angular Momentum

• Linear momentum is conserved if the net external
  force acting on the system is zero.
• Angular momentum is conserved if the net external
  torque acting on the system is zero.

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Angular momentum is conserved by changing the
angular velocity

• When the masses are brought in closer to the
  students body, his rotational velocity increases
  to compensate for the decrease in rotational
  inertia.
• He spins faster when the masses are held close to
  his body, and he spins more slowly when his arms
  are outstretched.

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Angular momentum is conserved by changing the
angular velocity

• The diver increases her rotational velocity by
  pulling into a tuck position, thus reducing her
  rotational inertia about her center of gravity.

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Keplers Second Law

• Keplers second law says that the radius line
  from the sun to the planet sweeps out equal areas
  in equal times.
• The planet moves faster in its elliptical orbit
  when it is nearer to the sun than when it is
  farther from the sun.

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Keplers Second Law

• This is due to conservation of angular momentum.
• The gravitational force acting on the planet
  produces no torque about an axis through the sun
  because the lever arm is zero the forces line
  of action passes through the sun.

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Keplers Second Law

• When the planet moves nearer to the sun, its
  rotational inertia about the sun decreases.
• To conserve angular momentum, the rotational
  velocity of the planet about the sun must
  increase.

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Angular momentum is a vector

• The direction of the rotational-velocity vector
  is given by the right-hand rule.
• The direction of the angular-momentum vector is
  the same as the rotational velocity.

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A student holds a spinning bicycle wheel while
sitting on a stool that is free to rotate. What
happens if the wheel is turned upside down?

• To conserve angular momentum, the original
  direction of the angular-momentum vector must be
  maintained.

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A student holds a spinning bicycle wheel while
sitting on a stool that is free to rotate. What
happens if the wheel is turned upside down?

• The angular momentum of the student and stool,
  Ls, adds to that of the (flipped) wheel, -Lw,
  to yield the direction and magnitude of the
  original angular momentum Lw.

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A student sits on a stool holding a bicycle wheel
with a rotational velocity of 5 rev/s about a
vertical axis. The rotational inertia of the
wheel is 2 kgm2 about its center and the
rotational inertia of the student and wheel and
platform about the rotational axis of the
platform is 6 kgm2. What is the initial angular
momentum of the system?

1. 10 kgm2/s upward
2. 25 kgm2/s downward
3. 25 kgm2/s upward
4. 50 kgm2/s downward

L I? (2 kgm2)(5 rev/s) 10 kgm2/s upward
from plane of wheel