In the arrangement shown in the figure, find the tensions in the rope and accelerations of the masses $m_{1}$ and $m_{2}$ and pulleys $P_{1}$ and $P_{2}$ when the system is set free to move. Assume the pulleys to be massless and strings are light and inextensible.

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Solution

Given,
In an arrangement shown in the fig.
Finding the tension in the rope and acceleration of the masses $m_{1}$ and $m_{2}$ and pulley.
$m_{1} < m_{2}$
Now,
$m_{2}$ will be go down word with an acceleration $a$ . Tension on both sides of the pulley will be same.
Let, it be $T$ let us consider forces towards mass $m_{1}$
Here downward force is
$= m g - T$
It is also equal to $m_{1} a$
Therefore,
$m_{1} g - T = m_{1} a \to (1)$
On the other side $m$ is moving upwards so upwards forces
$= T - m_{2} g$ it is also equal to $m_{2} a$ so
$T - m_{2} g = m_{2} a \to (2)$
From equation $(1) a n d (2)$
$m_{1} g - m_{2} g = m_{1} a + m_{2} a a = \frac{(m_{1} - m_{2}) g}{m_{1} + m_{2}}$
Also from equation $(1)$ and $(2)$
$T = \frac{2 m_{1} m_{2} g}{m_{1} + m_{2}}$

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Q. In the arrangement shown in figure,find the tensions in the rope and accelerations of the masses

m_{1}

and

m_{2}

and pulleys

p_{1}

and

p_{2}

when the system is set free to move. Assume the pulleys to be massless and strings are light and inextensible.

In the arrangement shown in figure find the tensions in the rope and accelerations of the masses $m_{1}$ and $m_{2}$ and pulleys $P_{1}$ and $P_{2}$ when the system is set free to move. Assume the pulleys to be massless and strings are light and inextensible.