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Question

In the pulley system shown in the figure the movable pulleys $$A, B$$ and $$C$$ are of mass $$1\ kg$$ each. $$D$$ and $$E$$ are fixed pulleys. The strings are light and inextensible. Find tension in the string.

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Solution

Since, all strings are connected to each other, it will have some tension at all point, let it be $$T$$.
Considering the acceleration of pulley,  $$A,B$$ and $$C$$ is $$a_1,a_2$$ and $$a_3$$ downward.
$$\rightarrow$$ according to constrain equation,
$$\sum T.a=0$$
$$-2Ta_3-T.a_2-Ta_1=0$$
$$2a_3+a_2+a_1=0\quad\rightarrow(1)$$
Applying Newton''s laws on each block,
$$10-2T=a_3\rightarrow(2)$$
$$10-T=a_2\rightarrow(3)$$
$$10-T=a_1\rightarrow(4)$$
From equation $$(2)$$ and $$(3)$$
$$-T=a_3-a_2$$
$$T=a_2-a_3 \rightarrow(5)$$
putting it in
$$10-a_2+a_3=a_2$$
$$2a_2=10+a_3\rightarrow(6)$$
Solving $$(2)$$ and $$(4)$$, we get,

$$a_1=a_2=\dfrac{10}3,\quad a_3=-\dfrac{10}{3}$$

$$T=\dfrac{20}3$$Newton

Physics

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