A long metal rod of length l and relative density σ is held vertically with its lower end just touching the surface of water. The speed of the rod when it just sinks in water is given by
√2gl(1−12σ)
Let the densities of metal and water be ρ and ρ0 respectively and let x be the length of the rod immersed in water at an instant of time t. Then, acceleration at that instant = apparent weight divided by mass of the rod, i.e.
dvdt=πr2lρg−πr2xρ0gπr2lρ=g−gxρ0lρ=g(1−xσl)
⇒dvdx.dxdt=g(1−xσl)
⇒vdvdx=g(1−xσl)
Integrating, we have
v22=g∣∣x−x22σl∣∣l0=gl(1−12σ)
⇒v=√2gl(1−12σ)
Hence, the correct choice is (c).