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Newton's Laws of Motion

Three blocks ...

Question

Three blocks of mass $m_{1}, m_{2}$ and $m_{3}$ are connected as shown in the figure. All the surfaces are frictionless and the string and the pulleys are light. Find the acceleration of $m_{1} .$

a_{0} = \frac{g}{1 + \frac{m_{1}}{4} (\frac{1}{m_{2}} + \frac{1}{m_{3}})}

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a_{0} = \frac{2 g}{1 + \frac{m_{1}}{4} (\frac{1}{m_{2}} + \frac{1}{m_{3}})}

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a_{0} = \frac{g}{1 + \frac{m_{1}}{4} (\frac{2}{m_{2}} + \frac{1}{m_{3}})}

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a_{0} = \frac{g}{1 + \frac{m_{1}}{4} (\frac{1}{m_{2}} + \frac{2}{m_{3}})}

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Solution

The correct option is A $a_{0} = \frac{g}{1 + \frac{m_{1}}{4} (\frac{1}{m_{2}} + \frac{1}{m_{3}})}$
Suppose the acceleration of $m_{1}$ is $a_{0}$ towards right.
The acceleration of pulley B will also be $a_{0}$ downwards because the string connecting $m_{1}$ and B is constant in length.
Also the string connecting $m_{2}$ and $m_{3}$ has a constant length.
This implies that the decrease in the separation between $m_{2}$ and and B equals the increase in the separation between $m_{3}$ and B. So, the upward acceleration of $m_{3}$ with respect to B.
Let this acceleration Be $a_{r} .$
The acceleration of $m_{2}$ with respect to the ground $= a_{0} - a_{r}$ (downward) and the acceleration of $m_{3}$ with respect to the ground $a_{0} + a_{r}$ (downward).
Let the tension be $T_{1}$ in the upper string and $T_{3}$ in the lower string.
Consider the motion of the pulley B.
The forces on this light pulley are
a) $T_{1}$ upwards by the upper string and
b) $2 T_{2}$ downwards by the lower string.
As the mass of the pulley is negligible, $2 T_{2} - T_{1} = 0$ $⟹ T_{2} = \frac{T_{1}}{2}$ ... (i)
Motion of $m_{1}$ : In the horizontal direction, the equation is
$T_{1} = m_{1} a_{0}$ ...(ii)
Motion of $m_{2}$ : $m_{2} g - T_{1} / 2 = m_{2} (a_{0} - a_{r})$ ...(iii)
Motion of $m_{3}$ : $m_{3} g - T_{1} / 2 = m_{3} (a_{0} + a_{r})$ ...(iv)
Solving these four equations, we get $a_{0} = \frac{g}{1 + \frac{m_{1}}{4} (\frac{1}{m_{2}} + \frac{1}{m_{3}})}$

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Three blocks of mass m1,m2 and m3 are connected as shown in the figure. All the surfaces are frictionless and the string and the pulleys are light. Find the acceleration of m1.

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