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 ((m)/(k))`
B. `2 pi[((TA +KL)m)/(LAK)]^(1//2)`
C. `2pi (mYA)/(KL)`
D. `2pi (mL)/(YL)`

Answer

B

Explanation

The wire may be treasted as a string for which force constant
`K = ("Force")/("Extension") = (YA)/(L) (:' Y = (F)/(A) xx (L)/(DeltaL))`
spring constant of the spring `k_(2) = 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)/(YAL)]^(1//2)`