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Pulley + Block + Rope

A uniform rope of linear mass density $\lambda $ is used to release block m with uniform acceleration a. Find the tension at a point P on the rope at a distance l from the block as shown in the figure.


Let mass m and the rope of length l above the block be the system.

Mass of rope of length $l = \lambda l$

For the system under consideration, $(m + \lambda l)g - {T_P} = (m + \lambda l)a$

$ \Rightarrow {T_P} = (m + \lambda l)(g - a)$