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) + 2pi sqrt((2mL)/(AY))`
C. `(2L)/(v) + pi sqrt((mL)/(AY))`
D. `(2L)/(v)`

Answer

A

Explanation

`K _(Q) = (YA)/(L)`
`K_(p) = ((2Y)(A//2))/(L) = (YA)/(L)`
`:. K_(p) = K_(Q) = K = (YA)/(L)`
Half oscillation is completed with `P` and half with `Q`. But their value of `K` is same. Hence, we can say that in one oscillation one time period is comleted with spring.
`T = ((2L)/(v)) + 2pi sqrt((m)/(K)) = (2L)/(v) + 2pi sqrt((mL)/(AY))`