question_answer

A horizontal uniform rope of length L, resting on a frictionless horizontal surface, is pulled at one end by force F. What is the tension in the rope in a distance l from the end where the force is applied?

A) (a)\[F\left( 1-\frac{l}{L} \right)\]  

B) \[2F\left( 1-\frac{l}{2L} \right)\]

C) \[\frac{F}{L}\]  

D) \[\frac{F}{l}\left( 1-\frac{l}{L} \right)\]

Correct Answer: A

Solution :

[a] Let a be the acceleration of the rope and M be its total mass. Then \[T=\frac{M}{L}\left( L-l \right)a\]   ?.(i) \[\text{and }F-T=\frac{M}{L}\times \ell a\]   ?(ii) Dividing (i) and (ii), \[\frac{F-T}{T}=\frac{\ell }{L-\ell }\Rightarrow F(L-\ell )-T(L-\ell )=T\ell \] \[\Rightarrow F(L-\ell )=T(L-\ell +\ell )=T\times L\] \[\Rightarrow T=F\left( 1+\frac{\ell }{L} \right)\]