Question

A uniform rope of length L lies on a table. If the coefficient of friction is μ, them find the maximum length x of the part of the rope, which can lie on the surface of the table without sliding down.

Answer 1

The maximum length x of the part of the rope is $x=\mu L-\mu x \Rightarrow x=\mu \frac{L}{1+\mu}$

Solution:

The uniform rope length = L  

Co-efficient of friction = $\mu$

So, the maximum length of the table without sliding can be calculated  

Weight of hanging part balanced by force of friction over table with part lying on the table.  

Mass per unit of length of the rope be $=\frac{M}{L}$

Thereby, the weight of hanging part be

$W=\left(\frac{M x}{L}\right) g$ (With x length hanging)

Weight to balance by Force of friction be f  

$f=\mu\left(M \frac{[L-x] g}{L}\right)$

$\Rightarrow W=f$

$\Rightarrow \frac{M x g}{L}=\mu\left(M \frac{[L-x] g}{L}\right)$

Solving this equation, we get,

$x=\mu L-\mu x \Rightarrow x=\frac{\mu L}{1+\mu}$

Answer 2

Answer:

mu*l/1+mu

hope above answer will help you