Falling Chain

A chain of mass M and length L is suspended vertically with its lower end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length x of the chain has fallen? Neglect the size of the individual links.

SOLUTION

The reading of the scale consists of two parts: the weight of the chain accumulated and the impulse per unit time imparted by the chain colliding with the scale. The first part is clearly W₁ = Mgx/L. The velocity of the links at the instant of hitting the scale is found from v² = 2gx. The second component of the force exerted on the scale equals

The total force on the scale during the fall of the chain is therefore