Question

In the system shown, mass of block A, B and C are $m_A=4m,m_B=3m$ and $m_C=8m$. If the system is released from rest, find the acceleration of block $B$. Assume all surfaces to be smooth and string inextensible.

• A. $\frac{g}{8}$
• B. $\frac{g}{2}$
• C. $\frac{g}{4}$
• D. $g$

Answer 1

The correct option is B $\frac{g}{2}$

Let the given system is reframed as shown
So, from the FBDs of $A,B$ and $C$, we get

$T=3m{a}_B$ ... (1)

In vertical direction: $4mg-T=4m{a}_A$ ... (2)
In horizontal direction: $N_2-N_1=4m{a}_C$ ... (3)
$N_1+T-N_2=8m{a}_C$ ... (4)

From constrained condition, using method of intercept we have

${\stackrel{..}{L}}_1+{\stackrel{..}{L}}_2=0\Rightarrow a_A-a_C-a_B=0$
$a_A=a_C+a_B$ ... (5)
Adding eq (2) and (3), we get,
$4mg-T+N_2-N_1=4m(a_A+a_C)$
Using eq. (4), we get
$4mg-8m{a}_C=4m(a_A+a_C)$
$\Rightarrow 4mg=4m(a_A+a_C+2a_C)$
$\Rightarrow a_A+3a_C=g$ ... (6)
From (3) & (4), we get
$T=12m{a}_C$ ... (7)
From (1) & (7) we have,
$3m{a}_B=12m{a}_C\Rightarrow a_B=4a_C$ ... (8)
From (5), (6) & (7) we have,
$g-3a_C=a_C+4a_C\Rightarrow a_C=\frac{g}{8}\Rightarrow a_B=\frac{g}{2}$