Derive a 2 order ODE from a physical problem

[lo1206]

Homework Statement

The motion of various shaped objects that bob in a pool of water can be modeled by a second order differential equation derived from F = ma.

The forces acting on the object are:
1)force due to gravity,
2)a frictional force by water,
3)and a buoyant force based on Archimedes' principle: An object that is completely or partially submerged in a fluid is acted on by an upward force equals to the weight of the water it displaces.

Derive the ODE by:
1)the dependent variable is the depth z of the object's lowest point in water,take z to be negative downward so that z= -1 means 1 ft of the object has submerged.
2)Let V(z) be the submerged volume of the object
3)m be the mass of the object
4)p be the density of the water
5)g be the acceleration due to gravity
6)w be the coef of friction for water(friction is proportional to the vertical velocity of the object)

Homework Equations

The Attempt at a Solution

I am not sure whether my answer is correct since I had confusion in treating positive and negative sign.

z''+ (w/m)z'-(pVg)/m +g =0

Thx very much for your attention

 

[chrisk]

With a force pointing up as positive the net force, F, acting on the object is

F = -F_f + F_b - F_g

So, is your sign convention correct?

 

[lo1206]

With a force pointing up as positive the net force, F, acting on the object is

F = -F_f + F_b - F_g

So, is your sign convention correct?

but I don't think that friction force must be negative since it opposes the direction of velocity,so I propose that

F= -wv + pVg - mg (negative sign for friction force since it opposes the velocity, pVg is positive since it points upward, mg is downward hence it is negative)

Does my argument make sense?

Moreover, V depends on the depth z, if we consider the volume of a cube of length L,
would it be V=L²z or V= -L²z ?
in case we define volume as positive

 

[chrisk]

Your statement about friction opposing the direction of velocity is correct. Since F_f = wv the minus sign for F_f provides the correct direction of the frictional force. Regarding the volume, it's part of the bouyant force expression. You can state that V(z) = V(-z) in the ODE and leave F_b as positive (upward force) in the force equation, or your can place a negative sign in front of the bouyant force expression if V(z) = -V(z) as you suggested.

 

[paranormal]

Your answer is close, but there are a few corrections that need to be made.

First, the dependent variable should be the position of the object, not the depth. This is because the object's position will change as it bobs up and down in the water, while the depth will remain constant.

Second, the buoyant force should be included in the equation. This force is equal to the weight of the water displaced by the object, which can be calculated using the submerged volume and the density of the water.

Finally, the frictional force should be proportional to the velocity, not the acceleration. This means that the coefficient of friction should be multiplied by z', the first derivative of z with respect to time.

With these corrections, the equation can be written as:

mz'' + wz' - pVg + mg - ρVgz' = 0

Where:
m = mass of the object
w = coefficient of friction for water
p = density of water
V = submerged volume of the object
g = acceleration due to gravity
ρ = density of the object

This is a second-order ordinary differential equation, which can be solved to determine the position of the object at any given time.