The floating cylinder

Consider a cylinder of length L and density r floating in a liquid of density s. Let the cylinder have a cross-sectional area A and let a length h be below the surface when the cylinder is at rest. (See Figure 1)


The cylinder is now pushed downwards a little (x) and allowed to bob up and down, the forces causing the oscillation being gravity and the varying upthrust on the cylinder.

Extra upthrust = extra weight of liquid displaced = Aσgx

Therefore restoring force = Aσgx = ma

Acceleration (a) = - Aσgx/m = - Aσgx/AρL = - [σg/ρL]x

The acceleration is therefore directly proportional to the displacement (x) and so the cylinder therefore moves with simple harmonic motion.

The value of ω² for this system is σg/ρL so the period T is:

Period of floating cylinder (T) = 2π/ω = 2π(h/g)^1/2

since for a floating body the upthrust = the weight of the body, that is, ALρ= Ahσ.