# Equation for a floating solid sphere

1. Jul 6, 2011

### Grand

1. The problem statement, all variables and given/known data
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?

2. Jul 6, 2011

### ehild

Yes, you have to include it.

ehild

3. Jul 6, 2011

### Grand

Any hint on how to do this?

4. Jul 6, 2011

### ehild

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

ehild

5. Jul 6, 2011

### 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?

6. Jul 6, 2011

### ehild

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

ehild

7. Jul 7, 2011

### 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.

8. Jul 7, 2011

### 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