In the given figure, two elastic rods P and Q are rigidly joined to end supports. A small mass m is moving with velocity v between the rods. All collisions are assumed to be elastic and the surface is given to be smooth. The time period of small mass m will be (A=area of cross section, Y=Young's modulus, L=length of each rod)

Options

A. `(2L)/v+2pisqrt((mL)/(AY))`
B. `(2L)/v+2pisqrt((2mL)/(AY))`
C. `(2L)/v+pisqrt((mL)/(AY))`
D. `(2L)/v`

Explanation

Rod behaves as spring constant `(YA)/l`
Equivalent system is

The time period of oscillations of block is
`T=(2L)/V+1/2(2pisqrt((mL)/(YA)))+1/2(2pisqrt((mL)/(2Y.A//2)))`
`=(2L)/V+2pi((mL)/(AY))`