Question

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

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

Answer 1

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

Given that,

Mass = $M$

Length = $L$

Tension = $T$

We know that,

$dT=dmg$

$\int dT=\underset{0}{\overset{L-l}{\int }}dmg$

Now,

$\frac{dm}{dx}=\frac{M}{L}$

$dm=\frac{M}{L}dx$

Now, put the value of dm

$T=\underset{0}{\overset{L-l}{\int }}\frac{M}{L}dxg$

$T=\frac{Mg}{L}[L-l]$

$T=(1-\frac{l}{L})Mg$

Hence, the tension is $\left(\frac{L-l}{L}\right)Mg$