How Does Archimedes's Law Apply to Harmonic Motion?

[Omri]

Hi everyone,
I have 2 small questions for you, both related to (simple) harmonic motion.

First, I have this diagram:
http://img227.imageshack.us/img227/8530/archimedesvu2.jpg
The mass of the cylinder is m, the area of a cut is S, and the water's density is p (supposed to be the Greek rho but p is the closest...). The questions says that according to Archimedes's law, the lift force on the cylinder equals the weight of the volume of the water filled by the cylinder. As this is my first encounter with this law, I tried to "build" the force using what I'm given. My first idea was that the mass of the water filled by the cylinder is p*S*h (h being the depth of the cylinder). I'm just not so sure about the dimensions - this is only good if the dimensions of p are kilograms divided by meters-cube. I'll be glad if anyone could show me the right way to formulate the force.

In the second question we have this diagram:
http://img468.imageshack.us/img468/9214/twospringsqn2.jpg
And the key question is - what is the equation of motion?
I think it's supposed to be a = -k1*x - k2*x , which is just like connecting the mass to a single spring with force constant k1+k2. Is this right?

Many many thanks!

 

[mgb_phys]

1, You have the right idea, the float receives a force equal to the weight of the water displaced, density (rho) is kg/m^3 and the volume displaced is just S * h
Since the force is directly proportional to the distance from equilibrium it's simply SHM just the same as a spring. The only complication is that when more of the float is above the water (ie it is bouncing up) then the restoring force is S * h * g.

2, Sounds correct, the second spring would act to make the first spring stronger and vice-versa.

 

[Omri]

Thank you very very much! :-)