Question

Find the tension in rope at section A, at a distance x from the right end.

Answer 1

If the mass of the rope is $M$, then acceleration,
$a=\dfrac{F}{M}$
Now, consider a portion of mass $m$ of the rope from its left end to $x$ distance from the right end.
If the linear mass density of the rope is assumed constant then,
$\lambda =\dfrac{M}{L}=\dfrac{m}{L-x}\\ \Rightarrow m=M\left(1-\dfrac{x}{L}\right)$
Now, considering this portion of rope,
$T=ma\\ =\dfrac{F}{M}\cdot M\left(1-\dfrac{x}{L}\right)\\ \boxed{T=F\left(1-\dfrac{x}{L}\right)}$