ROTATIONAL KINETIC ENERGY

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Session
Rotational Mechanics - 5
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Session Objective

1. Rolling of a body Description
2. Rolling of a body - Mathematical expression
3. Instantaneous center of rotation

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Rolling of a body
Rolling without slipping Motion is combined
rotation and translation.
Point of contact has no relative motion with
surface
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Rolling of a body
Friction (f) is responsible for rolling without
slipping
No work is done by f when body rolls without
slipping
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Relationship between linear angular velocity
The point in contact with surface is momentarily
at rest. Linear velocity is velocity of center
of mass (o).
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Relationship between linear angular velocity
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Dynamics of rolling
Rolling without slipping Linear motion of
center of mass circular motion about center of
mass.
Point P is at rest with respect to the surface at
this instant. It is the instantaneous center of
rotation .
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Kinetic Energy
When considered with respect to P
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Kinetic Energy with respect to instantaneous axis
of rotation
Velocity vQ of the point Q, with respect to the
instantaneous centre rotation
are valid about instantaneous point of rotation.
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Class Exercise - 1
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Solution
Ring I mr2
For same mass and KE, sphere has maximum
linearvelocity.
Hence, answer is (c).
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Class Exercise - 2
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Solution
For a sphere,
Hence, answer is (a).
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Class Exercise - 3
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Solution
Speed at B v
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Solution contd..
Hence, answer is (d).
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Class Exercise - 4
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Solution
Loss in PE Gain in KE
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Solution contd..
Hence, answer is (b).
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Class Exercise - 5
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Solution
Loss in PE Gain in KE when M has dropped a
height h.
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Solution contd..
Hence, answer is (c).
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Class Exercise - 6
A ring and a cylinder of the same mass m and the
same radius R are released with the same velocity
at the same time on a flat horizontal surface
such that they start rolling immediately on
release towards a wall equidistant from both. The
rolling friction is finite. Which of the two will
reach the wall first?
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Solution
For rolling,
As m and v are same, KE is same for both. For the
ring I mr2
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Solution contd..
So the cylinder will reach first.
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Class Exercise - 7
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Solution
As friction is negligible, Loss in KE Gain in PE
The body is probably a hollowsphere.
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Solution contd..
Hint/Formula
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Class exercise - 8
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Solution
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Class Exercise - 9
A disc of radius R is spin about its axis so that
its angular velocity is w. Then it is slowly
placed on a rough horizontal surface of
coefficient of friction m. How long will the disc
takes to stop? (Plane of the disc is parallel to
surface all through the rotation.)
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Solution
Take a concentric ring of radius r andwidth dr
(as one of a series ofconcentric rings making up
the disc).
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Solution contd..
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