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Question

A uniform rope of length $$L$$ is pulled by a force $$F$$ on a smooth surface. Find tension in the rope at a distance $$x$$ from the end where force is applied.


Solution

Let the mass of the rope $$AB$$ be $$m$$ . 
Acceleration of the rope = $$a$$ = $$\dfrac{F}{m}$$
Let the force be applied at the end A and the point at a distance $$x$$ from point $$A$$ be $$P$$ .
Tension in the rope at a distance x from end $$A$$ (at point $$P$$)
                    = Force required to move the remaining part with acceleration $$a$$
Linear mass density of the rod = $$\dfrac{m}{l}$$
Mass of the remaing part i.e. part $$PB$$ = $$\dfrac{m}{l}(l-x)$$
Force required to move the remaining part = mass*acceleration
                                                                         = $$\dfrac{m}{l}(l-x)$$$$\dfrac{F}{m}$$
Therefore , tension in the rope at a distance $$x$$ from the 
end at which the force is applied is  $$F(\dfrac{l-x}{l})$$.

Physics

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