Equation for a floating solid sphere

[Grand]

Homework Statement

A sphere is floating in water. It is pushed just under the water level and released. I'm asked to write the equation of motion for the sphere, not assuming small oscillations.

Is it just:
$$my''=\rho V g - mg$$
?

Or do I have to include that the buoyant force is changing while the volume under water changes as well?

 

[ehild]

Or do I have to include that the buoyant force is changing while the volume under water changes as well?

Yes, you have to include it.


ehild

 

[Grand]

Any hint on how to do this?

 

[ehild]

Calculate the volume of the sphere submerged in water as function of y.

ehild

 

[Grand]

I found that the submerged volume is:
$$V=\frac{4\pi}{3}a^3-\frac{\pi(a-y)^2}{3}(3a-(a-y))=\frac{\pi}{3}(2a^3+3a^2y-y^3)$$
but how do I find the period of oscillations from here on?

 

[ehild]

The text says that you have to write the equation of motion.
What is y?

ehild

 

[Grand]

The problem says find the eq of motion and then the period of oscillations (not small) but surely the first step is the equation of motion.

y is the coordinate of the centre of the sphere.

 

[ehild]

Are you familiar with non-linear second order differential equations? Can you find limit cycles? It is certainly not Introductory Physics.
You can find the period of small oscillation around the equilibrium position of the sphere.

ehild