Determine the period of the oscillations

[physixnot4me]

1) A long cylindrical rod of radius, r is weighted on one end so that it floats upright in a fluid having a density. It is pushed down a distance x from its equilibrium position and released. Show that the rod will execute simple harmonic motion if the resistive effects of the fluid are negligible and determine the period of the oscillations>>

Start with the forces: mg and the "restoring" force, bouyancy. Bouyance is dependent on displacement.

mg- restoring force = mass*acceleration

where acceleration is d"x/dx".

BUT, how does that lead me to showing the rod will execute simple harmonic motion? and how do i go about determining period of oscillations from there?

 

[durt]

show that acceleration is proportional to the negative of the displacement

 

[physixnot4me]

?

i'm not sure how to go about that.. show it is proportional to negative displacement? Are there are any steps for this type of mathematical writeup? any ideas or help would be appreciated. thanks again.

 

[Psi-String]

If you can prove the that$$F= -kx$$
where F is the resaultant force on the rod x is displacement and k is a constant, then this is a SHM.