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  4. A uniform rope of length L, resting on a frictionless horizontal surface, is pulled at one end by a force F. The tension in the rope at a distance I from this end is

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Q. A uniform rope of length $L$, resting on a frictionless horizontal surface, is pulled at one end by a force $F$. The tension in the rope at a distance $I$ from this end is

Punjab PMETPunjab PMET 2011Laws of Motion

Solution:

Mass of length $l=m L$, where $m$ is mass per unit length of the rope.
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From Newton's 2 nd law,
Force $=$ mass $\times$ acceleration
$F=m L a$
$\Rightarrow a=\frac{F}{m L}$
Tension in rope at distance $l$ from the given end
$T =m(L-l) a$
$=m(L-l) \frac{F}{m L}$
$T=\left(1-\frac{l}{L}\right) F$