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A pendulum made of a uniform wire of cross-sectional area A has time period T. When an additional mass M is added to its bob, the time period changes to Tm. If the Young's modulus of the material of the wire is Y then 1/Y is equal to (g = gravitational acceleration)

1. [(Tm/T)² - 1] A/(Mg)
2. [(Tm/T)² - 1] Mg/A
3. [1 - (Tm/T)²] A/(Mg)
4. [1 - (Tm/T)²] Mg/A

Correct answer: [(Tm/T)² - 1] A/(Mg)

Solution

The correct option is derived from the relationship between the time period of a pendulum and the properties of the material, where the change in time period due to the added mass can be expressed in terms of Young's modulus, cross-sectional area, and gravitational force. This option accurately reflects how the increase in mass affects the time period, leading to the derived formula.

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