| a. |
A
particle P moves at a constant angular speed w in a circular path of radius
A centered at the origin O. Sketch to show the net force on P, putting
down its linear speed and acceleration. No mathematical proofs are required. |
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The speed of P is
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The acceleration of P is the centripetal
acceleration for it to move in a circular path. a points towards O
and is |
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| b. |
At
time t = 0, the line joining P and the origin makes an angle f with the
y-axis. Q is the projection of P on the x-axis. Sketch a new diagram to
show the positions of P and Q after time t. Write down the mathematical
expressions for the displacement, velocity and acceleration of Q. Hence,
show that the motion of Q is simple harmonic. |
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marks |
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The equations for displacement, velocity and acceleration
are |
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From (3) and (5), |
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Thus, the motion of Q is simple harmonic. |
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| c. |
If
Q represents the location of a mass attached to a horizontal spring. Discuss
how the motion of the mass is started if
i)
f = 0,
ii)
f = p/2,
iii)
f = p,
iv)
f = 3p/2. |
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i) For f
= 0, the particle is projected from the equilibrium position to the increasing
direction of x. |
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ii) For f
= p/2, the particle is pulled to the right and
then released. |
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iii) For f
= p, the particle is projected from the equilibrium
position to the decreasing direction of x. |
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iv) For f
= 3p/2, the particle is pulled to the left and
then released. |
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| d. |
Discuss
how the energy of Q varies with
i)
the stiffness of the spring used, |
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marks |
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The energy of Q includes both kinetic energy
and potential energy. However, it is a constant which is equal to the maximum
of k.e. or maximum p.e. |
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Stiffness of the spring is proportional to
k.
Thus, energy of Q is proportional to stiffness of the spring: |
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0.5 |
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ii)
the amplitude of oscillation.
Use appropriate graphs to illustrate your answer. |
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The energy of Q is proportional to the square
of the amplitude: |
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