Question

One end of a long metallic wire of length L 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 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 given by

• A. $2\pi \sqrt{\frac{m}{k}}$
• B. $2\pi {\left[\frac{(YA+KL)m}{YAK}\right]}^{\frac{1}{2}}$
• C. $2\pi \left(\frac{mYA}{KL}\right)$
• D. $2\pi \left(\frac{mL}{YA}\right)$

Answer 1

The correct option is B $2\pi {\left[\frac{(YA+KL)m}{YAK}\right]}^{\frac{1}{2}}$

If the wire extends by an amount x when a force F is applied to it, then
$y=\frac{\frac{F}{A}}{\frac{x}{L}}=\frac{FL}{Ax}orF=\left(\frac{YA}{L}\right)X=kx;k_2=\frac{YA}{L}$
Thus the force constant of the wire is k. if K is the forece constant of the spring, then the force constant of the series combination of the wire and the spring is given by
$\frac{1}{k^{\prime }}=\frac{1}{k_1}+\frac{1}{k_2}$
$Or{k}^{\prime }=\frac{k_1{k}_2}{(k_1+k_2)}=.........\left(i\right)$
The time period of the combination is
$T^{\prime }=2\pi \sqrt{\frac{m}{k^{\prime }}}$
Using (i) in (ii) we find that the correct choice is (b)