Question

Consider a solid cylinder of the density `rho_(s)` cross section area A and h ploating in a liquid of density `rho_(l)` as shown in figure `(rho_(l) gt rho_(s))` . It is depressed sligtly and allowed to oscillation.

Answer 1

At equilibrium the net force on the cylinder is zero in the vertical direction.
`F_("net") = B - W = 0, B = ` the buoyancy and W = the weight of the cylinder direction.
When the cylinder is depresed slightly by x, the buoyancy increases from `B to B + sigma B`, where `sigma B = |x| rho_(l) Ag`
The equation of motion is, therefore.
`rho, Ah (d^(2)x)/(dt^(2) = - (rho_(l) g)/(rho_(s) h)`
and the angular frequency, `omega`, is
`omega = sqrt((g rho_(l))/(h rho_(s)))`