A chain of length l and mass m lies in a pile on the floor. If its end A is raised vertically at a constant speed v, express in terms of the length y of chain that is off the floor at any given instant (a) the magnitude of the force P applied to A, (b) the reaction of the floor.

mass per unit length = m / l

At the given instant, length of chain in air = y

mass of chain hanging in air = my / l

weight of chain = mgy / l

Let F be the force applied to pull the chain in upward direction

F - weight of chain = mass x acceleration

F - mg y / l = 0 (acceleration is zero)

F = mg y / l

b) Reaction of the floor = weight of chain lying on the floor

= m g - weight of chain in air

= mg - mgy / l

= mg (1 - y / l)