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Applying NLM along 3 Axis

In the figure...

Question

In the figure, the blocks $A, B,$ and $C$ of mass m each have acceleration $a_{1}, a_{2},$ and $a_{3}$ respectively. $F_{1}$ and $F_{2}$ are external forces of magnitude $2 m g$ and $m g$ respectively. Then

$a_{1} = a_{2} = a_{3}$
$a_{1} > a_{3} > a_{2}$
$a_{1} = a_{2}, a_{2} > a_{3}$
$a_{1} > a_{2}, a_{2} = a_{3}$

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Solution

The correct option is B
$a_{1} > a_{3} > a_{2}$

Step 1: Acceleration for Block $A$
Let $T_{A}$ be the tension in the string, $g$ be the acceleration due to gravity.
The equation of motion for mass $A$ is
$T_{A} - m g = m a_{1}$
From the given figure, the tension on mass $A$ $= 2 m g$ due to the downward acceleration due to gravity.
$\Rightarrow 2 m g - m g = m a_{1}$
$\Rightarrow a_{1} = g$
Step 2: Acceleration for Block $B$
Let $T_{B}$ be the tension in the string, $g$ be the acceleration due to gravity.
The equation of motion for mass $B$ is
$T_{B} - m g = (m + 2 m) a_{2}$
From the given figure, the tension on mass $B$ $= 2 m g$ due to the downward acceleration due to gravity.
$\Rightarrow 2 m g - m g = 3 m a_{2}$
$\Rightarrow a_{2} = \frac{g}{3}$
Step 3: Acceleration for Block $C$
Let $T_{C}$ be the tension in the string, $g$ be the acceleration due to gravity.
The equation of motion for mass $C$ is
$T_{C} - m g = (m + m) a_{3}$
From the given figure, the tension on mass $C$ $= 2 m g$ due to the downward acceleration due to gravity.
$\Rightarrow 2 m g - m g = 2 m a_{3}$
$\Rightarrow a_{3} = \frac{g}{2}$
Therefore $a_{1} > a_{3} > a_{2}$ .
Hence, the correct answer is option (B).

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