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| Period of a Simple Pendulum |
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| A simple pendulum is an arrangement in which a point mass is suspended by an inextensible weightless string in a uniform gravitational field. This is an ideal system which cannot be
realized in practice. However, a pendulum consisting of a small but relatively heavy bob on one end of a very light string can be considered as a simple pendulum. |
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| In its equilibrium position, the thread will be vertical with the weight mg of the bob being balanced by the tension in the string. If the bob is pulled aside and released, it oscillates about its equilibrium position. |
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| Let P1 represent the position of the bob when it is displaced through a small distance x. The component of mg tangential to the arc PP1 provides the restoring force. |
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| Applying the formula s = r q for the arc PP1, we get |
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| The motion is approximately simple harmonic. Comparing this equation with the standard equation from SHM, namely |
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| F = -Kx, we get |
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| The period in SHM is given by |
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| The period of a simple pendulum |
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 is directly proportional to the square root of the length |
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 is inversely proportional to the acceleration due to the gravity |
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 is independent of the mass of the bob |
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 is independent of the amplitude of oscillation, provided the amplitude is small |
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| Clocks are usually fitted with pendulums
which take one second to move from one end of their swing to the other end.
Since period is the time taken for a to and fro motion, the period of a
second's pendulum is 2 seconds. Let L represent the length of such a
pendulum. Then substituting in equation (13), we get |
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| Squaring and rearranging, we get |
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