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 the 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 :

Options

A. `2pisqrt((m)/(k))`
B. `2pisqrt((m(YA+kL))/(YAK))`
C. `2pisqrt((mYA)/(kL))`
D. `2pisqrt((mL)/(YA))`

Answer

B

Explanation

(b) Wire is equivalent to a spring of spring constant, `k'=(AY)/(L)`
k' and k are in series
`k_(eq)=(k'k)/(k'+k)=((AY//L)+k)/(((AY)/(L))+k)=(AYk)/(AY+kL)`
`T=2pisqrt((m)/(k_(eq)))=2pisqrt((m(AY+kL))/(AYk))`