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An
object of mass m attached to a rod is caused to move in a vertical circular
path of radius r at a constant speed v. Write down the forces inside the
rod at the highest position and at the lowest position. Hence, determine
the minimum value of v so that the force inside the rod is tension throughout
the motion. |
5
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Assume that the force inside the rod is always
tension. (If it proves to be compression, the value of force is negative.) |
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At the highest position, |
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At the lowest position, |
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Notice that if v is too small, T1
would be zero or negative in equation (1). Thus, for the rod to be in tension,
the minimum speed is given by setting T1 in equation
(1) to zero: |
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| b. |
An
object is tiled to one end of a string and is hung vertically. Discuss
the motion of the object if it is given with different initial speeds at
this position. |
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There are three possible motions: |
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In case (i), the speed of projection is less than required
to reach the centre. |
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In case (ii), the speed at the highest position is less
than that for T to be non-zero there. |
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i.e. the speed at the lowest position is given by |
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In case (iii), the speed at the highest position
is high enough for the string to remain taut. |
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| c. |
An
object is tiled to one end of a string. Draw force diagrams for the object
in the following motions:
i)
the object is set to move in a horizontal circle, describing a conical
pendulum. |
5
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ii)
the object is set to move in a vertical plane, describing simple pendulum.
Explain how the
tension in the simple pendulum varies with the position of the pendulum. |
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Since the pendulum bob is performing circular
motion with different speeds at different positions, the tension varies
with position. The difference of the tension and the component of weight
along the string gives the centripetal force required. |
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At the lowest point, the speed is highest.
The tension is the largest. |
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