Air resistance upon a spinning block

[Lonewolf723]

I came up with this problem, and solved for it, but I do question its accuracy, and id like to know how I could make it a better model. (it is not a homework problem)

So, a block, of dimensions a (rests on the z axis) by b (rests on the y axis) by c (rests on the x axis), is moving with a constant velocity v in the z direction and is spinning in the z-y plane about the center of mass that is located in the geometric center of the rectangular block with a angular frequency ω starting at time t from a angle of θ₀.The angle was measured from the y-axis to b, so when it is at θ₀ b is on the y-axis and a is on the z axis. It moves through space and encounters air resistance form air of density ρ. Also a*c is less than b*c

So the general equation for any drag is: F_D=(C_D*ρ*A*v²)/2 where A is the projected surface area in the z direction, C_D is the drag coefficient which depends upon the shape of the object and F_D is just the force encountered by the object from the fluid with a density ρ
The problem was finding A and C_D.

The projected area I using simple geometry to be c(b*cos(θ₀+ω*t)+a*sin(θ₀+ω*t)).

It was harder to find C_D as it varied with the angle so I decided to make a approximation. I made it a function of the angle, making it a cosine function with α being the maximum and β the minimum value of the drag coefficient. SO my equation for the coefficient would be:
C_D=((α-β)*cos(θ₀-ω*t)+α+β)/2
As far as I know this is supposed to be a cosine function, but I'm not 100 percent positive.

The I just input the two equations into the original F_D equation giving me a quite big equation which I shall not show as its annoyingly large, but unnecessary to show as you can just figure it out.

My main question is how can I make C_D a more realistic function and how accurate my approximation is from known reality. Also if anyone knows, how would the angular slow down due to the air resistance and if there is some sort of study done on such a topic.

 

[256bits]

From your descrption I infer that your block is doing sumersaults like gymnastic at the Olympics.

If so, depending upon the ratio of v to ω - ie if ω is large in realtion to v, then you just have a spinnig block, or, if ω is very small in relation to v then you just have a block moving through the air where the projected area slowly changes.

Cd is determined by experiment in wind tunnel tests for different shapes and varies with the Reynold's Number.
So, if you have a reference on how Cd varies with the angle of the projected shape please provide, as I would be most interested.

 

[Lonewolf723]

Well the point of the model was to determine the air resistance for a block for which the v I not much bigger than ω.

Also about Cd, it would take a minimum value when the block was at a angle of 90 or 270 while having a maximum value when it is at 0 or 180, as that is the time when the maximum surface area projected. So the biggest question with how to get a function of Cd is to find how it changes with angle, which I suppose has to be found experimentally. Anyone know of any research papers done on this topic or one similar to it? From what I figured Cd(min) would be around 0.8 while Cd(max) would be around 2.4.

But yeah 256bits, I wonder what the equation would be for air resistance for a block with a very high ω. I suppose it would not be as highly variable as in my original model. This might be non-sense, but could one make a approximation that the block would have a Cd and A equal to that of a sphere? Maybe only for extremely high ω.

Lastly, anyone know the equation of the air resistance upon a spinning block that is not moving through space, but is simply rotating and experiencing only perpendicular force on the moving side. Now that I come to think of it, I think I got a idea how to get it but It seems odd.

So, for the block the air resistance would take the form of the classical air resistance formula of Fd. Cd would be constant, as would ρ. Now, v would be different for each section for the block, and taking the center of the block to be the origin and setting the long side it to be parallel to the x axis, r would be the distance of the velocity is from the center then: v=rω, as is the classical formula.

So now I am not sure I did the correctly, but I just took A to be c*b, so I then I had to sum up the force upon the block. so I integrated the function with respect to r from -b/2 till b/2. The result was: ΔF_D=(C_D*ρ*ω²*c*b⁴)/2

Now I am not sure this is correct, but I guess it might, as it makes sense that it is so dependent upon b, as for example a very long rod it stops to spin very quickly, as the drag imposes a large torque upon the rod. Now that I come to think of it... I would have solved for torque then deduced the total force or something. eh anyone can help?