One end of a long mettalic wire of length L is tied to the ceiling. The other end is tied to 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 T equal to

Options

A. `2pi ((m)/(K))`
B. `2 pi {((YA + KL)m)/(YAK)}^(1//2)`
C. `2 pi (m YA)/(KL)`
D. `2 pi (mL)/(YA)`

Answer

B

Explanation

The wire may be treated as a spring for which force constant
`k_(1) = ("force")/("extension") = (YA)/(L) " "(Y = (F)/(A) xx (L)/(Delta L))`
spring constant of the spring `k_(1) = K`
Hence spring constant of the combination (series)
`k_(eq) = (k_(1)k_(2))/(k_(1) + k_(2)) = (((YA)/(L))K)/(((YA)/(L))+K) = (YAK)/(YA + KL)`
Time period :
`T = 2pi sqrt((m)/(k)) = 2pi [((YA + KL)m)/(YAK)]^(1//2)`