Question

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to mass less spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross-section and 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

The wire may be treated as a string for which force constant
$k_1=\frac{Force}{Extension}=\frac{YA}{L}(\because Y=\frac{F}{A}\times \frac{L}{\Delta L})$
Spring constant of the spring $k_2=K$
Hence spring constant of the combination (series)
$k_{eq}=\frac{k_1{k}_2}{k_1+k_2}=\frac{\left(YA/L\right)K)}{\left(YA/L\right)+K}=\frac{YAK}{YA+KL}$
$\because$ Time period $T=2\pi \sqrt{\frac{m}{k}}=2\pi {\left[\frac{(YA+KL)m}{YAK}\right]}^{\frac{1}{2}}$