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Intro to Torque

In the arrang...

Question

In the arrangement of figure shown above, the masses $m_{0}$ , $m_{1}$ and $m_{2}$ of bodies are equal, the masses of the pulley and the threads are negligible, and there is no friction in the pulley. Find the acceleration $\to w$ with which the body $m_{0}$ comes down, and the tension of the thread binding together the bodies $m_{1}$ and $m_{2}$ , if the coefficient of friction between these bodies and the horizontal surface is equal to $k$ . Consider possible cases.

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Solution

Let us write the fundamental equation of dynamics for all the three blocks in terms of projections, having taken the positive direction of x and y axes as shown in fiigure below and using the fact that kinematical relation between the accelerations is such that the blocks move with same value of acceleration (say $w$ )
$m_{0} g - T_{1} = m_{0} w$ ......(1)
$T_{1} - T_{2} - k m_{1} g = m_{1} w$ ......(2)
and $T_{2} - k m_{2} g = m_{2} w$ ......(3)
The simultaneous solution of equations (1), (2) and (3) yields:
$w = g \frac{[m_{0} - k (m_{1} + m_{2})]}{m_{0} + m_{1} + m_{2}}$
$T_{2} = \frac{(1 + k) m_{0}}{m_{0} + m_{1} + m_{2}} m_{2} g$
As the block $m_{0}$ moves down with acceleration $w$ , so in vector form:
$\to w = \frac{[m_{0} - k (m_{1} + m_{2})] \to g}{m_{0} + m_{1} + m_{2}}$

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