  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. 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,$$a_1=a_2=\dfrac{10}3,\quad a_3=-\dfrac{10}{3}$$$$T=\dfrac{20}3$$NewtonPhysics

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