Q: One end of a long metallic wire of length L_{0} is tied to the ceiling. The other end is tied to a massless spring of spring constant k. A mass m hangs freely from the free end of the spring. The area of cross-section and the Young’s modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to

(a) $ \displaystyle 2\pi \sqrt{\frac{m}{k}}$

(b) $ \displaystyle 2\pi \sqrt{\frac{m Y A}{k L_0}}$

(c) $ \displaystyle \pi \sqrt{\frac{m(YA + kL_0)}{k Y A}}$

(d) $\displaystyle 2\pi \sqrt{\frac{m(YA + kL_0)}{k Y A}}$

Ans: (d)