In the figure shown mass `2m` is at rest and in equilibrium. A particle of mass `m` is released from height `(4.5mg)/(k)` from plate. The particle sticks to the plate. Neglecting the duration of collision. Starting from the when the particles sticks to plate to the time when the spring is in maximum compression for the first time is `2pisqrt((m)/(ak))` then find `a`.

Answer

3

Explanation

Velocity of the narticle just before collision
`u = sqrt(2g xx (4.5 mg)/(K))`
`u = 3gsqrt((m)/(K))`
Now it collides with the plate.
Now just after collision velocity of systeam of plate `+` particle
`mu = 3mv`
`rArr V = (u)/(3) = gsqrt((m)/(K))`
Now system perform 's `SHM` with time period `T = 2pisqrt((3m)/(K))`and mean position as `(mg)/(K)` distance below the point of collision.
Let the equation of motion be.
`y = A sin(omegat + phi)`
for `t = 0 y = ng//K`
`(mg)/(K) = Asinphi ....(1)`
Now for amplitude
`V = omegasqrt(A^(2) - x^(2))`
`gsqrt((m)/(K)) = sqrt((K)/(3m)) sqrt(A^(2) - (m^(2)g^(2))/(K^(2))) , (sqrt(3)(mg)/(K))^(2) = A^(2) -(m^(2)g^(2))/(K^(2))`
`A = (2mg)/(K) ....(2)`
By `(1)` & `(2)`
`T = 2pisqrt((3m)/(K))`
`X = (A)/(2)` to `x = 0 rArr t = (T)/(12)`
`x = 0` to `x = A rArr t = T//4`
total time `= (T)/(12) + (T)/(4) = (2pi)/(3)sqrt((3m)/(K)) = 2pisqrt((m)/(3k)) :. a = 3`