motion of CM

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Motion in Nature

• motion of CM
• motion around CM

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Motion of a Rigid Body (Solid Object)

• motion of CM
• rotational motion around CM

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Uniform Circular Motion
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Angular Velocity and Angular Acceleration
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Angular
Linear
Related by
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Constant a relations Constant ?
relations
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?
M1
r1
M2
Note the radius is Determined by the size Of the
circle traced out.
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Rotational Kinetic Energy

• Rotational inertia (or Moment of inertia)

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For extended bodies, it is more complicated
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homogeneous body
y
R
x
If rotational symmetry about Z axis
R
z
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Sphere centered on (0,0,0) find moment of
inertia for rotation about z axis.
R
y
x
y
R
x
a
R
z
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Moment of Inertia for solid sphere about its
center of mass cont.
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P 253
It is complicated, thats why we have tables!
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Parallel Axis Theorem
h
Relate moment of inertia about axis through
COM to a new axis parallel to it but displaced by
distance h.
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Proof of parallel axis theorem
Consider the a set of mass points with center of
mass(0,0,0) Find the moment of inertia about
the axis parallel to the z axis and passing
through (a,b,0)
Since the center of mass is (0,0,0)
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For the diehard same proof for extended body
Consider the moment of inertia for an object
about an axis parallel to the z axis and passing
through (a,b,0) (where (0,0,0) is the CM)
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Applying the Parallel Axis Theorem
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D2cm
Applying the Parallel Axis Theorem
RB15 cm
MB3kg
MR1kg
h for either ball
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Mass point on rigid rotor
?
m
r
Axis
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Torque
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Linear to Angular Relations
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Constant a relations Constant ?
relations
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Rigid Body About Fixed Axis
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Newtons Second Law for Rotation
Work done by torque
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Angular Momentum
Angular momentum and Torque
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Conservation of Angular Momentum
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In a tuck, rapid rotation
Opened out, slow rotation
COM follows a parabola
opened out Again for entry
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Angular
Linear
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• Angular position
• Angular displacement

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Rolling
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Kinetic Energy of Rolling
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Rolling without slipping down an incline
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Massive Pulleys
m2
That means one fewer Equation, so use 2nd law for
sum of torques.
m1
Use one extra condition
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m2
m1
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