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.

Answer 1

Let the mass of the rope $AB$ be $m$ .

Acceleration of the rope = $a$ = $\frac{F}{m}$

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 = $\frac{m}{l}$

Mass of the remaing part i.e. part $PB$ = $\frac{m}{l}(l-x)$

Force required to move the remaining part = mass*acceleration

= $\frac{m}{l}(l-x)$$\frac{F}{m}$

Therefore , tension in the rope at a distance $x$ from the

end at which the force is applied is $F\left(\frac{l-x}{l}\right)$.