How can the natural frequency of a stationary cylinder be determined?

[Ry122]

In this problem I guess I would first need to model the position of the cylinder with an equation and from that solve for the natural frequency. But how do I go about doing this when the object's default position is stationary?

 

[gabbagabbahey]

What is the force on the cylinder when it is displaced from its equilibrium position by a distance y (up or down)?

 

[Ry122]

It is equal to the weight of the water displaced I believe.
But how can I form an equation of motion from this?

 

[gabbagabbahey]

It is equal to the weight of the water displaced I believe.

There are two forces acting on the cylinder at any given time:

(1) The downward force of gravity

(2) The buoyancy force (pressure) exerted by the water

Find a mathematical expression for the net force on the cylinder when it is displaced from equilibrium by a distance y

But how can I form an equation of motion from this?

Newton's 2nd Law maybe?