Question

The uniform chain of mass m and length l hangs on a thread and touches the surface of a table by its lower end.Find the force exerted by the table on the chain when half of its length has fallen on the table. The fallen part does not from heap.

Answer 1

Consider a link falls from $h$ height , then stops

$F_{stop}=\frac{dP}{dt}=\frac{dm}{dt}v$

The velocity of that link will be $v_f=\sqrt{2gh}$

Here, the mass $dm$ is the mass of the elemental length $dx$ of chain

So,

$dm=\left(\frac{M}{L}\right)dx\Rightarrow \frac{dm}{dt}=\frac{M}{L}\frac{dx}{dt}\Rightarrow \frac{dm}{dt}=\frac{M}{L}v$

Now,

$F_{stop}=\frac{M}{L}\times 2gh$

At the time our link hits a fraction $\frac{h}{L}$ of chain is on the table. So, its weight is $Mg\frac{h}{L}$

Therefore,

$F_{stop}=2\times Weight$