Question

One end of a long metallic wire of length L is tied to the celling. 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.
• B.
• C.
• D.

Answer 1

The correct option is B

From the definition of Young's modulus Y, we find K of wire to be $=\frac{YA}{L}$

The combination of the wire and the spring can be visualised as the series combination of two springs of spring constants, $K_1$ and $K_2$.

$K_{eq}=\frac{K_1{K}_2}{K_1+K_2}=\frac{\frac{YAK}{L}}{\frac{YA}{L}+K}=\frac{YAK}{YA+LK}$

$\therefore T=2\pi \sqrt{\frac{m}{K_{eq}}}=2\pi \sqrt{\frac{m(YA+LK)}{YAK}}$