First slide

Connected bodies

Question

In the system shown in figure $m_{B}$ = 4kg, and $m_{A}$ = 2 kg. The pulleys are massless and friction is absent everyrwhere. The acceleration of block A is

Moderate

A

$\frac{10}{3}$ $m / s^{2}$
B

$\frac{20}{3}$ $m / s^{2}$
C

2 $m / s^{2}$
D

4 $m / s^{2}$

Solution

For movable pulley
$a_{B} = \frac{O + a_{A}}{2}$
$a_{A} = 2 a_{B}$ --------(i)
Free body diagrams of A and B:

Equation of motion of block A and B
For A : $T - \frac{m_{A} g}{2} = m_{A} a_{A} = m_{A} (2 a_{B}) - - - - - (ii)$
For B : $m_{B} g - 2 T = m_{B} a_{B}$ -------------(iii)
From (ii) and (iii) [2(ii) + (iii)]
$m_{B} g - m_{A} g = [4 m_{A} + m_{B}] a_{B}$
$\Rightarrow a_{B} = \frac{[m_{B} - m_{A}] g}{[4 m_{A} + m_{B}]} = \frac{[4 - 2] \times 10}{[4 \times 2 + 4]}$
= $\frac{20}{12} = \frac{5}{3} m / s^{2}$
Hence $a_{A} = 2 a_{B} = \frac{10}{3} m / s^{2}$

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