How to Derive the ODE for a Floating Spherical Float?

[yoyo]

I am trying to set up a first order differential equation to express the oscillation of a spherical float floating on water. I know that the two forces acting on the spherical float is force buoyancy and gravity and there is two density (density of water and density of the sphere). So first what i did was let y(t)= the top of the sphere [0, L]

this is where I'm stuck. I know that I am suppose to find an expression for submerged volume and use Newton's eqaution to derive the differential equation but having difficulties with that...please help..

 

[Nate810]

One approach is to consider the total energy of the system, which can be written as the sum of the potential energy (due to gravity) and the kinetic energy. The kinetic energy can be expressed in terms of the velocity of the float, and the potential energy can be expressed in terms of the displacement of the float from its equilibrium position. You can then use Newton's second law to write a first-order differential equation for the displacement of the float.

 

[thepatientmental]

To set up the first-order differential equation for the oscillation of a spherical float floating on water, we need to consider the forces acting on the float. As you mentioned, the two main forces are buoyancy and gravity.

We can start by defining the variables involved in the problem. Let h(t) be the height of the float at time t, and let r be the radius of the float. The volume of the float can be expressed as V = (4/3)πr^3.

Now, we need to determine the submerged volume of the float. This is the volume of water that is displaced by the float and is equal to the volume of the float that is below the water surface. We can express this as Vsub = (4/3)πr^3 - πr^2h.

Using Archimedes' principle, we know that the buoyant force acting on the float is equal to the weight of the water displaced by the float. This can be expressed as Fb = ρwaterVsubg, where ρwater is the density of water and g is the acceleration due to gravity.

The gravitational force acting on the float is simply Fg = mg, where m is the mass of the float and g is the acceleration due to gravity.

Now, we can set up Newton's second law of motion, which states that the sum of the forces acting on an object is equal to its mass times its acceleration. In this case, the acceleration is given by the second derivative of the height, so we have:

m(d^2h/dt^2) = Fb - Fg

Substituting the expressions for Fb and Fg, we get:

m(d^2h/dt^2) = ρwaterg((4/3)πr^3 - πr^2h) - mg

Simplifying, we get the first-order differential equation:

(d^2h/dt^2) + (ρwaterg/m)((4/3)πr^3 - πr^2h) = g

This equation can be solved using various methods, such as separation of variables or integrating factors, to determine the oscillation of the float.

I hope this helps you in setting up the differential equation for the buoyancy problem. Good luck!