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 is equal to : (g = gravitational acceleration)

Engineering

Physics

Stress Strain and Hookes Law New

Question

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 T_M. If the Young's modulus of the material of the wire is Y then $\frac{1}{Y}$ is equal to : (g = gravitational acceleration)

$[1 - {(\frac{T_{M}}{T})}^{2}] \frac{A}{Mg}$

$[1 - {(\frac{T}{T_{M}})}^{2}] \frac{A}{Mg}$

$[{(\frac{T_{M}}{T})}^{2} - 1] \frac{A}{Mg}$

$[{(\frac{T_{M}}{T})}^{2} - 1] \frac{Mg}{A}$

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$T = 2 π \sqrt{\frac{ℓ}{g}} \frac{Mg}{A} = \frac{YΔℓ}{ℓ}$

$T_{M} = 2 π \sqrt{\frac{ℓ + \frac{Mg}{AY} ℓ}{g}} Δℓ = \frac{Mgℓ}{AY}$

$\frac{T_{M}}{T} = \sqrt{(\frac{Mg}{AY} + 1)}$

$(\frac{T_{M}^{2}}{T^{2}} - 1) \frac{A}{Mg} = \frac{1}{Y}$