$4\mathrm{.}20$ Figure P$4\mathrm{.}20$ illustrates a cylindrical buoy floating in water with a mass

density $.$ Assume that the center of mass of the buoy is deep enough so that

the buoy motion is primarily vertical. The buoy mass is $m$ and the diameter

is $D.$ Archimedes' principle states that the buoyancy force acting on a floating

object equals the weight of the liquid displaced by the object. $($a$)$ Derive the

equation of motion in terms of the variable $x,$ which is the displacement from

the equilibrium position. $($b$)$ Obtain the expression for the buoy's natural

frequency. $($c$)$ Compute the period of oscillation if the buoy diameter is $4$ ft

and the buoy weighs $2000$ lb $.$ Take the mass density of fresh water to be

$=1\mathrm{.}94$ slug $\frac{?}{f{t}^3}.$

Figure P$4\mathrm{.}20$