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 and a mass m is hung from the free end of the spring. If m is slightly pulled down and released, then its time period of oscillation is

Options

A. `2pisqrt(m/k)`
B. `2pisqrt((mYA)/(kL))`
C. `2pisqrt((m(kA+YL))/(kYA))`
D. `2pisqrt((m(kL+YA))/(kYA))`

Answer

D

Explanation

For oscillating mass at end of a rod. Restoring force
= `(YA)/L*x`

So, `k_(1)` = spring constant for a rod is `(YA)/L`.
If a rod spring are connected, then it is a series combination.
So, `(k_(eq))` system = ` (k_(1)k_(2))/(k_(1)+k_(2))`
= `=(KYA//L)/(k+(YA)/L)=(kYA)/(kL+YA)`
So, `T=2pisqrt(m/k_(eq))rArrT=2pisqrt((m(kL+YA))/(kYA))`