A block with mass (M) is connected by a massless spring with stiffness constant (k) to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position (x_0). Consider two cases : (i) when the block is at (x_0) , and (ii) when the block is at `x = x_0 + A`. In both the cases, a particle with mass m(lt M) is softly placed on the block after which they strick to each other. Which of the following statement (s) is (are) true about the motion after the mass (m) is placed on the mass (M) ?

Options

A. The amplitude of oscillation in the first case changes by a factor of `sqrt(M)/(m +M)`, whereas in the second case it remains unchanged.
B. The final time period of oscillation in both the cases is same.
C. The total energy decreases in both the cases.
D. The instantaneous speed at (x_0) of the combined masses decreases in both the cases.

Answer

A, B, C

Explanation

Case (i) : Applying conversation of linear momentum.
`MV_1 = (M + m)V_2` -(1) :. `V_2 = ((M)/(M + m)) V_1`
(##JMA_CHMO_C10_023_S01##),
`M(A_1 xx omega_1) = (M + m)(A_2 xx omega_2)`
:. `MA_1 xx sqrt (K)/(M) = (M + m) A_2 xx sqrt (K)/(M + m)`
:. `A_2 = sqrt(M)/(M + m) = A_1`
Also `E_1 = (1)/(2) MV_1^2`
and `E_2 = (1)/(2) (M + m)V_2^2 = (1)/(2) (M + m)`
`xx (M^2 V_1^2)/(M + m)^2) = (1)/(2) ((m)/(M + m))^2 V_1^2`
Clearly `E_2 lt E_1`
The new time period `T_2 = 2sqrt(m + M)/(k)`
Case (ii) : The new time period `T_2 = 2sqrt (m + M)/(K)`
Also `A_2 = A_1`
Here `E_2 = E_1`
The instantaneous value of speed at (X_o) of the combined masses decrease in both the cases.