Problem $2.$ Yeah Buoy!

A rectangular ocean buoy, of width and thickness b$,$ has been designed to wildly oscillate

at a critical wind speed. The buoy consists of a rectangular volume with an attached fan.

The fan oscillates at a frequency fan proportional to the incoming wind speed Vwind given

by fan $=$ Vwind

L $,$ where L is the length of the fan blade. An eccentric mass me is attached

a distance r from the center of rotation of the fan. As the fan rotates, due to the eccentric

mass me$,$ it provides a sinusoidal force in the y direction of the buoy.

Assume that the buoy is constrained to move only in the y direction and that the system

damping force can be represented by Fdamping $=$ c dy

dt $,$ where c $=1\mathrm{.}5$ N$$s

m $.$ The remaining

parameter values are as follows m $=100$ kg$,$ me $=1$ kg$,$ g $=10$ m

s$2,$ L $=1$ m$,$ r $=1$ m$,$

b $=1$ m$,$ and water $=1000$ kg

m$3.$ At equilibrium, the buoyant force balances the gravitational

force exactly and the buoy remains at rest. As the buoy is displaced, the buoyant force is

equal to the volume of the fluid displaced multiplied by the acceleration of gravity g and the

density of the fluid water$.$

$($a$)$ Provide and box the expression for the force Ffan that the fan exerts on

the buoy in the y$-$direction due to the rotating eccentric mass.