Setting up a differential equation (buoyancy)

1. Nov 22, 2014

24karatbear

1. The problem statement, all variables and given/known data
When an object is submersed in a liquid, it experiences a buoyant force equal to the weight of the liquid displaced by the object. As an object moves through a liquid, there is a resistive force which is directly proportional to the density of the liquid, the cross sectional area A of the object (perpendicular to the direction of motion) and the square of the speed v of the object. A spherical object of mass m and density λ > 1000 begins to sink in a pool of water of depth D. Set up the differential equation with initial condition for the depth of the object below the surface of the water. Use 1000 kilograms per cubic meter as the density of water.

2. Relevant equations
N/A

3. The attempt at a solution
I am choosing the downward direction (y-direction) to be positive. The object starts at the origin and descends to a depth D. We consider three forces, all in the y-direction: the weight of the spherical object, the buoyant force (= weight of the water displaced by the sphere), and the resistive force. I use Newton's 2nd law:
ΣFy = may = mobject(d2D/dt2) = Fobject - Fbuoyant - Fresistive = mobjectg - (1000 kg/m3)(4*πr3/3)g - kλπr2(dD/dt)2, (where k is just a constant of proportionality) and my initial condition would be D(0) = 0

Would this be correct?

edit: Added g for the buoyant force.

Last edited: Nov 22, 2014
2. Nov 22, 2014

SteamKing

Staff Emeritus
Using D as the depth of the pool and as the vertical coordinate can be confusing. Since you have already mentioned the forces acting on the sphere are in the y direction, why not use y as the general depth coordinate?

3. Nov 22, 2014

24karatbear

Oh, I was trying to be consistent with the variables given in the problem. Would it be incorrect to use D as I used it above?

4. Nov 22, 2014

SteamKing

Staff Emeritus
Like I said, it can get confusing using D as the depth of the bottom of the pool (a constant) and D as a variable quantity (the distance as measured from the surface of the pool). It's better to use an arbitrary variable to represent the position of the sphere w.r.t. the surface of the pool (x, y, or, z, for example)

5. Nov 22, 2014

24karatbear

Ah okay, I got it! I'll switch it out then.

Here's what I get after cleaning the equation up:

mobjg - (1000)(4*πr3/3)g - kλπr2(dy/dt)2 - mobj(d2y/dt2) = 0
--> (d2y/dt2) - g + (1000)(4*πr3/3)g/mobj + kλπr2(dy/dt)2/mobj = 0 (divided everything by -mobj)
--> (d2y/dt2) + kπr2λ(dy/dt)2/mobj + g[(1000)(4*πr3/3)/mobj -1] = 0, Initial condition: y(0) = 0

Am I on the right track?

6. Nov 22, 2014

SteamKing

Staff Emeritus
You're getting warmer. Since this is a second order equation, you'll need two separate initial conditions to determine a unique solution. You have specified the initial starting position of the sphere. Can you think of another condition to specify?

7. Nov 22, 2014

24karatbear

Oh, right! I am guessing that I'll need y'(0) = 0? The problem doesn't say that it starts at rest, but I suppose I can assume that it does for the 2nd condition (feel free to correct me if I'm wrong).

8. Nov 22, 2014

SteamKing

Staff Emeritus
That would be a reasonable initial condition.