Question

A uniform rope of length $L$ and mass $M$ is hanging from a rigid support. The tension in the rope at a distance $x$ from the rigid support is

• A. $Mg$
• B. $Mg\left(\frac{L-x}{L}\right)$
• C. $Mg\left(\frac{L}{L-x}\right)$
• D. $Mg\left(\frac{x}{L}\right)$

Answer 1

The correct option is B $Mg\left(\frac{L-x}{L}\right)$


Since the rope is uniform, mass per unit length for the rope remain same at each section.
Therefore $\frac{M}{L}=\frac{m_{L-x}}{(L-x)}$
$\Rightarrow m_{L-x}=M\left(\frac{L-x}{L}\right)$
From the FBD,

$T=\left(m_{L-x}\right)g=M\left(\frac{L-x}{L}\right)g$
Hence, tension in the rope at a distance $x$ from the rigid support $T=M\left(\frac{L-x}{L}\right)g$