Question

A rope of length $r$ placed on a frictionless horizontal surface is pulled horizontally by a force $F$ applied at one end. The tensile force $T$ developed in the rope at a distance $x$ from the rear end varies with $x$ as shown in the graph. Then it could be concluded about linear density (mass per unit length) of rope that:

• A. It is inversely proportional to $x$
• B. It is not proportional to $x$
• C. It increases with $x$
• D. It decreases with $x$

Answer 1

Answer: (C)

It increases with $x$

Now $M={\int }_0^{L}k{x}^n\cdot dx$ $M=\frac{k\cdot L^{n+1}}{n+1}$ $\begin{array}{cc}\text{Similarly,}& M_1={\int }_0^{x}k\cdot x^n\cdot dx\\ & =\frac{k\cdot x^{n+1}}{n+1}\\ \therefore T=& \frac{k\cdot x^{n+1}}{n+1}\times \frac{F(n+1)}{k\cdot L^{n+1}}\alpha x^{n+1}\end{array}$ $\therefore n=1$ (usiequation-----1)

$\text{Linear density increases linearly with X}$