A pendulum is made of a massless string of length \( L \) and a small bob of negligible size and mass \( m \). It is released making an angle \( \theta_0 \) (<< 1 rad) from the vertical. When passing through the vertical, the string slips a bit from the pivot so that its length increases by a small amount \( \delta \) (\( \delta \ll L \)) in negligible time. If it swings up to angle \( \theta_1 \) on the other side before starting to swing back, then to a good approximation, which of the following expressions is correct?
A pendulum is made of a massless string of length \( L \) and a small bob of negligible size and mass \( m \). It is released making an angle \( \theta_0 \) (<< 1 rad) from the vertical. When passing through the vertical, the string slips a bit from the pivot so that its length increases by a small amount \( \delta \) (\( \delta \ll L \)) in negligible time. If it swings up to angle \( \theta_1 \) on the other side before starting to swing back, then to a good approximation, which of the following expressions is correct?
Show Hint
- \( \theta_1 = \theta_0 \)
- \( \theta_1 = \theta_0 \left( 1 - \frac{\delta}{L} \right) \)
- \( \theta_1 = \theta_0 \left( 1 - \frac{\delta}{L} \right)^2 \)
- \( \theta_1 = \theta_0 \left( 1 - \frac{3 \delta}{2L} \right) \)
The Correct Option is D
Solution and Explanation
When the length of the pendulum increases by a small amount \( \delta \), the restoring force changes, and the angle \( \theta_1 \) reached after the swing will depend on this change. To a good approximation, the new angle is related to the initial angle by \( \theta_1 = \theta_0 \left( 1 - \frac{3 \delta}{2L} \right) \), accounting for the impact of both the length increase and the reduced amplitude of the swing.
Step 2: Conclusion.
Therefore, the correct answer is option (D).
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