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.
• B.
• C.
• D.

Answer 1

The correct option is D

Force constant of wire $k_1=\frac{F}{l}=\frac{YA}{L}$ and force constant of spring $k_2$ = k (given)
Equivalent force constant for given combination $\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}=\frac{L}{YA}+\frac{1}{k}$ $\Rightarrow$ $\frac{1}{k_{eq}}=\frac{kYA}{kL+YA}$
$\therefore$ Time period of combination $T=2\pi \sqrt{\frac{m}{k_{eq}}}=2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$