Let the mass of the whole chain be m, and the mass of unit length of the chain be m₀ = m/l. Let a part of the chain of length x be hanging down from the table at any moment; to begin with, x = l/2, where l is the length of the whole chain. Then the force which causes the chain to move will be proportional to the length of chain hanging down, i.e, m₀xg. The acceleration of the chain will equal m₀xg/m. To begin with, when m₀x = m/2 the acceleration equals g/2, but then it increases so that the chain's motion is not uniformly accelerated.
If identical masses M be attached to the ends of the chain, then at the moment when a part of the chain of length x is hanging from the table, the value of the moving force will be m₀xg + Mg = (m₀x + M)g, and the chain's acceleration at that moment will be (m₀x + M)g/(m + 2M). To solve the problem as to which case will cause the chain to slip off faster, we must find in which case will the acceleration increase at the greater rate. For this we must compare the two expressions:
m₀xg/m and (m₀x + M)g/(m + 2M)
To compare these fractions, we shall put them over a common denominator and compare their numerators. The numerator of the first fraction will be m₀xg(m + 2M) and that of the second fraction will be m₀xgm + Mgm.
It is clear that these numerators are equal at the beginning when m₀x = m/2. In the subsequent moments, the acceleration will be greater in the first case than in the second.
So the chain will slip off more quickly when there are no masses attached to its ends.
This result can also be obtained from the law of conservation of energy, taking into account that at the moment when the end of the chain slips off the table, its centre of gravity is at a distance of l/2 below the edge of the table. Then if v₁ be the velocity at that moment and if there are no masses attached to the end,
mg(l/2) = mv₁²/2
(Potential Energy) = (Kinetic Energy)
or gl = v₁².
while if masses are attached to the ends
(m + 2M)g(l/2) = (m + 2M)v₂²/2
or
gl = (m + 2M)v₂²/(m + M)
i.e, the time of slipping for the chain must be less when there are no masses attached to the ends.