In the arrangement of Fig. 1.9 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 m0 comes 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 Consider possible cases.

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 Fig; and using the fact that kinematical relation between the accelerations is such that the blocks move with same value 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 Eqs. (1), (2) and (3) yields,  $$w=g\frac{[{{m}_{0}}-k({{m}_{1}}+{{m}_{2}})]}{{{m}_{0}}+{{m}_{1}}+{{m}_{2}}}$$ and $${{T}_{2}}=\frac{(1+k){{m}_{0}}}{{{m}_{0}}+{{m}_{1}}+{{m}_{2}}}{{m}_{2}}g$$.

As the block m0 moves down with acceleration w, so  in vector form  $$\vec{w}=\frac{[{{m}_{0}}-k({{m}_{1}}+{{m}_{2}})]}{{{m}_{0}}+{{m}_{1}}+{{m}_{2}}}\vec{g}$$.

## Mechanics!

#### A point traversed half the distance with a velocity v0. The remaining part of the distance was covered with velocity v1 for half the time, and with velocity v2 for the other half of the time

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