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A
rotating system consists of N particles located at different positions
from the axis of rotation. Write down an expression for the moment of inertia
I of the system. Hence, discuss the factors that determine the value of
I for a rigid body. |
3
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Consider a system consists of N particles.
Suppose the mass of the ith particle is mi and
at a distance of ri from the rotating axis. The moment
of inertia of the system is |
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The factors that affects the value of I
are: |
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the mass of the body
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the way in which the masse is distributed (or shape), and
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the position of the axis of rotation
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| b. |
State
the Newton’s Second Law for rotation. Explain how the motion of a rigid
body acted on by two equal but opposite forces depends on the points of
application. |
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Newton's Second Law for Rotation |
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The torque acting on a rotating system is equal to the
time rate of change of angular momentum of that body. |
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Mathematically,
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For a rigid body (I does not change), |
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The separation between the two forces determines the size
of the torque. |
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The larger is the separation, the greater is the turning
effect. |
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| c. |
Describe
an experiment to determine the moment of inertia of a flywheel, stating
the measurements needed to be taken and how they can be integrated to give
the final result. |
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Step 1. Measure the radius, r, of the
axle of the flywheel with a caliper. |
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Step 2. Hang the mass with a string. Wind up
the mass by the axle through height h. |
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Step 3. Put a chalk mark on the flywheel for
counting no. of turns. |
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Step 4. Release the mass and count the no.
of turns N before the ground is reached. |
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Step 5. Further count the no. of turns N'
after the mass hits the ground.
Measure this time interval t by a stop watch. |
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As m falls, its gravitational p.e. is
converted into translational k.e., rotational k.e. and work done against
friction. |
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Let the energy loss per turn be Ef. |
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After hitting the ground, the k.e. of the flywheel
is dissipated in N' turns: |
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As it performs N' turns in time t,
we have |
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Combining (5), (6) and (7), it is possible
to find the moment of inertia I of the flywheel. |
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| d. |
Make
an analogy between rotational motion and rectilinear motion in terms of
i)
inertia |
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mass vs moment of inertia |
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ii)
Newton’s Second Law |
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Rectilinear |
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The rate of change of momentum of a mass is in the same direction
and directly proportional to the net force.
For a constant mass,
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Rotational |
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The torque is equal to the rate of change of angular momentum
of a rigid body.
For a rigid body,
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iii)
conservation law of momentum, and |
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Rectilinear |
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The linear momentum of a system is conserved if there is no
external force acting on the system.
For two masses involved in a collision,
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Rotational |
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The angular momentum of a system is conserved if there is no
external torque acting on the system.
For a rotating body whose shape changes,
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iv)
kinetic energy |
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Rectilinear |
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The kinetic energy of a mass m moving with speed v
is
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Rotational |
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The rotational kinetic energy of a system with moment of inertia
I rotating at angular speed w about an
axis is
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