Rotation ?? (Chap. 10)

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Rotation ?? (Chap. 10)

1. We are going to consider the rotation of a rigid
   body (a body of fixed shape and size) about a
   fixed axis.

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To begin, we have to define the angular position

1. When the body rotates, the angular position
   ?changes as a function of t.
2. ?plays the role of linear position x.

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Similarly, we have angular displacement ??
angular velocity???
with T the period and f the frequency.
also

• note that the angular velocity is also a vector,
  i.e. it has a direction (however, we only
  consider rotation about a fixed axis, so ? has
  only and sign.
• the direction is given by the right-hand rule.

angular acceleration????
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Is angular displacement a vector? The answer is
no because the addition of two angular
displacements depends on the order, which is
different from usual vector addition.
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Constant angular acceleration

1. for the case of constant angular acceleration,
   the equation of motion is equivalent to the
   equations for constant linear acceleration.
2. the derivation is identical to those in linear
   motion.

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Relation between linear and angular motion
The point P in the rotating body will follow a
circular motion.
the first derivative gives
the period the point P is
in other words,
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second derivation of s
Be careful that this value only present the
acceleration tangent to the circle.
there is another component of the acceleration
which is along the radius, i.e. the centripetal
acceleration (or radial acceleration) ar
the net acceleration is then
with
Note that at is non-zero only when there is an
angular acceleration while ar is non-zero even if
angular velocity is constant.
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Kinetic energy of rotating body
One can consider rotating body as a collection of
particles, and the kinetic energy is then simply
the sum of all kinetic energies of the particles
Note that all particles have the same angular
velocity.
Hence we can define a rotational inertia ???? I

• I (also called the moment of inertia ??? ) is a
  measure of its resistance to change in its
  angular velocity.
• This tells us how the mass is distributed about
  its axis of rotation.
• It is a constant for a particular rigid body and
  for a particular axis.

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And now the kinetic energy is given by
This is similar to the kinetic energy of linear
motion

1. The rotational inertia now plays the role of
   mass.
2. But it also related to the mass distribution the
   axis.

e.g. to rotate a rod about the central axis is
more easier than to rotate if perpendicular to
its length, it means the rotational inertia in
(a) is less than that in (b).
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Which one has largest rotational inertia?
The rotational inertia depends on the rotational
axis.
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Find the moment of inertia of the simple molecule
along A, B, C and D axes.
What is the rotational inertia about the axes A
and B?
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For a continuous body, the rotational inertia can
be calculated as
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For a thin rod with a uniform linear mass
density?M/L
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For a disc with density sM/(pR2)
How to calculate I about an axis at the edge of
the disc?
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Parallel-axis theorem
Proof Choose the origin locate at the center of
mass
Expanding the equation
0
0
(center of mass)
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I about an axis at the edge of the disc
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1. (a) How can one tell the difference between a
   had-boiled egg and a raw egg by trying to spin
   them? (b) After they are spinning, explain what
   happens if each is stopped for an instant and
   then released.
2. About which axis is the moment of inertia (a) the
   largest (b) the smallest?

1. Two identical cans of concentrated orange juice
   are released at the top of an incline. One is
   frozen and the other has defrosted. Which reaches
   the bottom first?

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Torque ??,??

1. A Force acting on a rigid body can be decomposed
   into two components one (Fr) is along the vector
   r and the other (Ft) perpendicular to r.
2. Fr does not cause rotation, only Ft will cause
   rotations about O. The ability of Ft to cause
   rotations also depends on the distance r. So we
   define the quantity torque as

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1. Torque, comes from the Latin word meaning to
   twist.
2. The unit of torque is Nm (Newton - meter), but
   note that it is different from the work which
   also has the unit NmJ. But the torque is never a
   measure of energy.
3. Same as forces, torques can adds together the
   total is the net torque of the body.

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Newtons second law for rotation
since
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Example
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The force needed is twice the force in (a).
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Work and rotational kinetic energy
Very similar to the case of linear motion
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