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. `2pi sqrt((m)/(K))`
B. `2pi sqrt((mYA)/(KL))`
C. `2pi sqrt((mK)/(KL))`
D. `2pi sqrt((m(KL + YA))/(KLA))`

Answer

D

Explanation

`K_(eq) = (K_(1)K_(2))/(K_(1) + K_(2)) = (K. (YA)/(L))/(K + (YA)/(L)) = (KYA)/(KL + YA)`
`T = 2pi sqrt((m)/(k_(eq))) = 2pi sqrt((m(KL + YA))/(KYA))`
(`because` string & spring in series)