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Pendulum clocks may run slow in summer a...

Pendulum clocks may run slow in summer and fast in winter .Why?

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We know that time period of pendulum clock is directly proportional to square root of length of pendulum . During summer , the length of pendulum increases with rise in temperature .So time period increases and clock runs slow and in winter ,length of pendulum decreases so time period decreases and clock runs fast .
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