Question

One end of a long metallic wire of length L, area of cross-section A and Young’s modulus Y is tied to the ceiling. The other end is tied to a massless spring of force constant k. A mass m hangs freely from the free end of the spring. It is slightly pulled down and released. Its time period is given by

• A. $2\pi\sqrt{\frac{m}{K}}$
• B. $2\pi\sqrt{\frac{mYA}{KL}}$
• C. $2\pi\sqrt{\frac{mK}{YA}}$
• D. $2\pi\sqrt{\frac{m(KL + YA)}{KYA}}$

Answer 1

Answer: (D)

$2\pi\sqrt{\frac{m(KL + YA)}{KYA}}$

Answer 2

Force constant of wire $\mathbf{k}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{F}}{\mathbf{l}}\mathbf{=}\frac{\mathbf{YA}}{\mathbf{L}}$ and force constant of spring $k_{2} = k$ (given)

Equivalent force constant for given combination

$\frac{1}{k_{eq}} = \frac{1}{k_{1}} + \frac{1}{k_{2}} = \frac{L}{YA} + \frac{1}{k}$ ⇒ $k_{eq} = \frac{kYA}{kL + YA}$

∴ Time period of combination

$T = 2\pi\sqrt{\frac{m}{k_{eq}}} = 2\pi\sqrt{\frac{m(kL + YA)}{kYA}}$