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

Correct answer: [(TM/T)² - 1] A/(Mg)

Solution

Adding mass M stretches the wire by dL = MgL/(AY), so T_M^2/T^2 = (L+dL)/L = 1 + Mg/(AY). Hence Mg/(AY) = (T_M/T)^2 - 1, giving 1/Y = [(T_M/T)^2 - 1]*A/(Mg).

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