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. ` 2 pi ((m)/(k))`
B. `2 pi { ((YA + kL)m)/(Yak)}^(1//2)`
C. `(2 pi)/( (mYA)/(kL)`
D. `2 pi (mL)/( YA)`

Answer

B

Explanation

Force constant for wire
`k_(1) = ("Force ") /("Extension") = (YA)/(L) ` [`:. Y = (F)/(A) xx (L)/( Delta L)` ]
Spring constant of the spring
`k_(2) = k`
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) = 2 pi sqrt(( m)/(k)) = 2 pi [ (( YA + kL )m)/(YAk)]^((1)/(2))`