| a. |
Derive
an expression for the work done by a torque acting on a rigid body which
is pivoted freely at a point. |
3
marks |
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Suppose the torque on the rigid body is created
by a force F at a distance r from the axis of rotation as
shown above. |
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As the object rotates through an angle q,
the force moves through an arc length |
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The work done by the force is |
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| b. |
Discuss
how a ballet dancer could spin faster suddenly. Account for the change
in kinetic energy of the dancer. |
4
marks |
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When a ballet dancer withdraws her arms, her moment of
inertia would decrease. |
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By conservation of angular momentum,
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As I decreases, the angular speed w
increases. |
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The rotational k.e. of the dancer is |
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Thus, the kinetic energy increases. |
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The dancer has to do work by withdrawing her arms. The
work done is transferred to her rotational k.e. |
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| c. |
Discuss
the force acting on a wheel which rolls at a constant speed on a horizontal
road. |
2
marks |
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Since the wheel is rotating at a constant speed,
by Newton's 1st Law, there is no external force acting on the wheel. |
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Because there is no external force, the torque
is zero. Thus, the wheel rolls with a constant angular speed. |
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| d. |
Explain
why a wheel rolling down a rough inclined plane takes a longer time than
the wheel sliding down a smooth plane of the same inclination. |
4
marks |
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By falling through the same vertical distance,
the loss in gravitational p.e. in the two cases are the same. |
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When a wheel rolls down, it gains in both translational
and rotational k.e. |
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Thus, the translational k.e. of a rolling wheel
is less than that of a sliding one. |
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The final speed of the rolling wheel is smaller.
Thus, the average speed is also smaller and it takes a longer time to finish
the journey. |
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| e. |
A solid
cylinder, a solid sphere and a hollow sphere are released simultaneously
from the top of a rough inclined plane. Comparing their linear speeds when
they reach the bottom of the incline. |
3
marks |
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The speed v of an object rolling down
an inclined plane through a vertical distance h is given by |
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It is clear that for a larger I, the speed
v is smaller. |
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Since the moment of inertia of cylinder, solid
sphere and hollow sphere are arranged in the following order: |
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Thus, the speeds reaching the bottom of the
incline follow: |
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