Setting up a differential equation (buoyancy)

[24karatbear]

Homework Statement

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.

Homework Equations

N/A

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:
ΣF_y = ma_y = m_object(d²D/dt²) = F_object - F_buoyant - F_resistive = m_objectg - (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.

 

[SteamKing]

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?

 

[24karatbear]

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?

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?

 

[SteamKing]

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)

 

[24karatbear]

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)

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

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

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

Am I on the right track?

 

[SteamKing]

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?

 

[24karatbear]

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?

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).

 

[SteamKing]

That would be a reasonable initial condition.