Question

One end of a long metallic wire of length L is tied to 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 Young's modulus of wire are A and Y, respectively. If the mass is slightly pulled down and released, it will oscillate with time period T equal to.

• A. $2\pi \frac{m}{K}$
• B. $2\pi {\frac{m(YA+KL)}{YAK}}^{1/2}$
• C. $2\pi \frac{mYA}{KL}$
• D. $2\pi \frac{mL}{YA}$

Answer 1

Answer: (A)

$2\pi \frac{m}{K}$

The wire may be treated as a starting 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)

$k_{eq}=\frac{k_1{k}_2}{k_1+k_2}=\frac{\left(YA/L\right)K}{\left(YA/L\right)+k}=\frac{YAK}{YA+KL}$

$\because TimeperiodT=2\pi \sqrt{\frac{m}{k}}=2\pi$ $\left[2\pi {\frac{m(YA+KL)}{YAK}}^{1/2}\right]$