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Question

In the arrangement of figure shown above, the masses m0, m1 and m2 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 w with which the body m0comes down, and the tension of the thread binding together the bodies m1 and m2, 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)
m0gT1=m0w ......(1)
T1T2km1g=m1w ......(2)
and T2km2g=m2w ......(3)
The simultaneous solution of equations (1), (2) and (3) yields:
w=g[m0k(m1+m2)]m0+m1+m2
T2=(1+k)m0m0+m1+m2m2g
As the block m0 moves down with acceleration w, so in vector form:
w=[m0k(m1+m2)]gm0+m1+m2
126886_129725_ans.png

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