Question
Why does the pendulum clock run slow in summer and fast in winter?

Answer

As we know that time period of pendulum clock $T =2 \pi \sqrt{\frac{ L }{ g }}$ at a place $g$ and $2 \pi$ are constant. so $T \alpha \sqrt{ L }$
The length of pendulum in summer increases so the time period of a pendulum is increased. So pendulum takes more time to complete one oscillation and the clock becomes slow in summer. Vice - versa in winter it becomes fast as a time period of a pendulum is decreased due to a decrease in length of a pendulum.

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