## Tuesday, May 13, 2008

### Irodov Problem 1.157

Let the mass per unit length of the falling chain be . The mass of an infinitesimally piece of chain of length dl is then given by. The momentum imparted by a piece of chain that is dl long and that hits the table with a speed v is given by
.

If this infinitesimally small piece of chain of length dl falls in time dt, the instantaneous force imparted by it (force is change in momentum per time) on the table at a speed v is given by,
. After a time t has elapsed since the chain starts to fall, the velocity of the tip of the chain that is hitting the table will be v=gt since it has been falling under the influence of gravity. This means that the force exerted by the falling chain as a function of time t will be .

In time t since the chain starts to fall, a length of would have fallen on the table (since it is freely falling under gravity). This means that after a time t has elapsed since the chain starts to fall the mass that lies on the table would be . The force exerted by this part of the chain would then be . Clearly this is half of the force exetrted by the falling chain.