In the adjoining figure, block A is of mass (m) and block B is of mass 2m. The spring has force constant k. All the surfaces are smooth and the system is released form rest with spring unstretched. Find the maximum extension in the spring and acceleration of block B at time of maximum extension
.

Answer

A, B, C, D

Explanation

(a) At maximum extension in the spring
`v_(A) = v_(B) =0` Therefore, appluing conservation of mechanical energy:
(momentarilt)
Therefore, appluing conservation of mechanical energy:
decreasing in gravitational energy of block B = incrasing in elastic potential
enerrgy of spring .
or `m_(B)gx_(M) =1/2Kx_(m)^(2)`
or `2mgx_(m) =1/2Kx_(m)^(2)`
:. `x_(m) =(4mg)/K`
(b) At `x=x_(m)/2 =(2mg)/K`
Let `v_(A) =v_(B) =v(say)`
The decrase in gravitational potential energy of block (B) =increase in elastic potential energy of spring + increase in kinetic inergy of both the blocks.
`:. m_(B) gx =1/2Kx^(2) +1/2(m_(A) + m_(B)) v^(2)`
or `(2m) (g) ((2mg)/k) = 1/2K ((2mg)/K)^(2) + 1/2(m+2m)v_(2)`
`:. v = 2gsqrtm/(2K)`
(c) At `x=x_(m/4)=(mg)/K`
.
or `Kx=mg`
`a=(Net pulling force)/(Total mass) =(2mg-mg)/3m`
` =g/3`(downwards).