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)//Ak)`
C. `2pisqrt(mYA//kL)`
D. `2pisqrt(mL//YA)`

Answer

B

Explanation

Stretching force, `F=YADeltaL//L`. The long metallic wire has the force constatn,
`k_(1)=(F)/(DeltaL)=(YA)/(L)`
Net force constant, as the string and spring are in series.
`K^(')=(kk_(1))/(k+k_(1))=(k(YA//L))/(k+YA//L)=(kYA)/(kL+YA)`
`:. T=2pisqrt((m)/(k^(')))=2pisqrt((m(YA+kL))/(kYA))`