A pendulum made of a uniform wire of cross-sectional area A...

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) $\quad[\mathbf{2 0 1 5}]$ (A) $\left[\left(\frac{T_{M}}{T}\right)^{2}-1\right] \frac{M g}{A}$ (B) $\left[1-\left(\frac{T_{M}}{T}\right)^{2}\right] \frac{A}{M g}$ (C) $\left[1-\left(\frac{T}{T_{M}}\right)^{2}\right] \frac{A}{M g}$ (D) $\left[\left(\frac{T_{M}}{T}\right)^{2}-1\right] \frac{A}{M g}$

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) $\quad[\mathbf{2 0 1 5}]$
(A) $\left[\left(\frac{T_{M}}{T}\right)^{2}-1\right] \frac{M g}{A}$
(B) $\left[1-\left(\frac{T_{M}}{T}\right)^{2}\right] \frac{A}{M g}$
(C) $\left[1-\left(\frac{T}{T_{M}}\right)^{2}\right] \frac{A}{M g}$
(D) $\left[\left(\frac{T_{M}}{T}\right)^{2}-1\right] \frac{A}{M g}$

Key Concepts

Pendulum Period and Effective Length

The period of a simple pendulum is determined by its effective length, with the period being proportional to the square root of the length. When a pendulum's support or suspension element is elastic, its actual length can change under varying loads, thereby altering the effective length and, consequently, the time period of oscillation.

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio between stress (force per unit area) and strain (relative change in length) in the elastic region. It quantifies how much a material will deform under a given load and is intrinsic to the material regardless of its shape or size.

Hooke's Law

Hooke's law states that, for small deformations, the extension or compression of a material is directly proportional to the applied force. This law is fundamental in understanding the elastic behavior of materials, serving as the basis for calculating displacements and changes in dimensions when a force is applied.

Elastic Extension in Suspended Systems

In systems where the supporting element (such as a wire or string) can stretch under additional weight, the extension of the element must be accounted for. The elastic extension affects the system’s dynamics by effectively increasing the length of the pendulum, which in turn modifies the period of oscillation as described by both elastic theory and pendulum motion equations.

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