Sol To solve this problem, we use the principle of conservation of energy. Let us denote by
y = 0 the plane . It will be the reference plane of the potential energy. In changing its position from A to B, the chain's potential energy changes, and as a result, its velocity changes. The potential energy is calculated by integration over the length of the chain. The mass of the chain per unit length is λ = . The contribution of a piece of length dr to the potential energy is

du = –gh dm = – λ hgdr …...(i)

Where h = r sin θ (see figure). The initial potential energy is, therefore,

u i = …...(ii)

Similarly, the final potential energy is :

u f = …...(iii)

From the principle of conservation of energy, we know that:

E k(i) + u i = E k(f) + u f …...(iv)

Where E k(i) = 0 Hence,

E k(f) = u i – u f = gsin θ (L 2 – b 2 ) mv 2 …...(v)

From which follows

v = …. (iv)