Question

The system shown in the figure is in equilibrium. Find the accelerations of the blocks A, B and C just after the string between A and B is cut. All blocks are of equal masses ‘$m$’ each and springs are of equal stiffness. (Assume springs to be ideal and take downward acceleration to be positive).

• A. $0,+2g,+g$
• B. $0,+g,+g$
• C. $-2g,+2g,0$
• D. $-2g,+2g,+g$

Answer 1

The correct option is C $-2g,+2g,0$

Since A and B are connected by inextensible string, therefore both will experience same acceleration. Hence they will be considered as single system.
Let the extension in the spring between ceiling and block A be $x_1$ and that for spring between B and C be $x_2$
Before cutting



For block C,
$\begin{array}{}K{x}_2=mg& \text{(1)}\end{array}$
For block A and B,
$K{x}_1=K{x}_2+2mg$
Since $K{x}_2=mg$
$\begin{array}{}\Rightarrow K{x}_1=3mg& \text{(2)}\end{array}$

When string between A and B is cut,



Let $a_A$, $a_B$ and $a_c$ be the acceleration of the block A, B and C after the string between A and B is cut.
For A:
$K{x}_1-mg=m{a}_A$
$3mg-mg=m{a}_A$ $\left[\text{Using}\right(2\left)\right]$
$\Rightarrow a_A=2g\uparrow$
For B:
$K{x}_2+mg=m{a}_B$
$\Rightarrow a_B=\frac{2mg}{m}=2g\downarrow$ $\left[\text{Using}\right(1\left)\right]$
For C:
$K{x}_2-mg=m{a}_c\Rightarrow a_c=0$
(as $K{x}_2=mg)$