Question

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 $l$ from this end is

• A. $F$
• B. $\frac{l}{L}F$
• C. $\frac{L}{l}F$
• D. $\left( 1-\frac{l}{L} \right)F$

Answer 1

Answer: (D)

$\left( 1-\frac{l}{L} \right)F$

Answer 2

Suppose $m$ be the mass of the rope per unit length. Therefore, total mass $>$ of length $L=m L$ $\therefore$ Force applied $F=(m L) a$ or $a=\frac{F}{m L}$ So, tension in the rope at a distance $l$ $=$ Force on the length $(L-0)$ or $T=m(L-l) a=m(1-l) \frac{F}{m L}$ or $T=\left(1-\frac{l}{L}\right) F$