Conservation Of Angular Momentum (COAM)

[gordonjudd]

Are you using a different approach other than L=rxp to get your pi/2 difference?

Walter,
Since you are getting a difference for the AM of a semi-circular loop measured in a moving frame vs the value calculated in an earth frame I am left to assume that you are not calculating AM with a straight forward cross product calculation.

If you could share more details on your method maybe we could sort out why we are getting different results.

Gordy

[Walter]

Gordy,

I agree. It would be nice to have consensus.

Unfortunately, when I did my calculations it was one of those “can’t sleep at night so let’s do calculus” sessions and I didn’t save the paper I was working on. I also got lazy and used one of those online integral calculators. I don’t really trust machines.

The angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv. Using the center of rotation that equates to the center of mass will give you an approximation for the rotation vector. It was how I came up with the first estimate of 2. In that calculation I also simplified the velocity field around the loop to decrease linearly.

Of course the velocity field around the loop doesn’t decrease linearly. It can be derived using the cycloid equation or to use the transform that you showed earlier (add a constant to the x velocity). Since it’s linear it will be the same in any Cartesian coordinate system as long as the orientation of the axes doesn’t change, i.e. it’s okay to change the origin.

The rotation vector is another issue since the center of the loop in the loop centric frame and at the bottom of the loop in the fixed frame.it’s magnitude does not remain constant in that view.

[gordonjudd]

The rotation vector is another issue since the center of the loop in the loop centric frame and at the bottom of the loop in the fixed frame.

Walter,
I think Noel picked the bottom of the loop for the instant center in his velocity calculation so that he would get a line velocity of 2*v_loop at the top of the loop and zero at the bottom. His eqn 1 will also fall out of using an instant center at the center of the loop and then adding the vo/2 value to the x velocity values calculated in a moving frame.

Using the center of rotation that equates to the center of mass will give you an approximation for the rotation vector.

I expect that you can pick any point for doing the AM calculation, but I don't think it makes sense to use the center of mass for that point since it does not fit with the value you would get using AM=omega*MOI calculation. Omega is constant if you use the center of the semi-circle in the omega=v_tan/r calculation. That would not be the case if you used the center of mass as the rotation center.

Unfortunately, when I did my calculations it was one of those “can’t sleep at night so let’s do calculus” sessions

If you can't repeat it that is too bad. But based on the results obtained with the much simplier L=rxp calculation I think your "calculus" formulation was flawed.

Gordy

[Walter]

Gordy,

When I said center of mass I meant center of rotation. In an rxp calculation in 2 d space the the cross product of 2 vectors a and b is equal to the magnitude of vector a multiplied by the magnitude of vector b multiplied by sin of angle between them multiplied by the unit vector n. I presume that mathcad is taking care of the magnitude and angle calculations for you when doing the cross product. I’m not sure what you think I was doing differently.

Since we were not in consensus I took a totally different approach and did an fea. You’ll be happy to know that in the case of the tethered cast I have the same result as you do.

The am of the loop is the same in the fixed and loop centric frames in a tethered cast.

I do get a different result when looking at shooting line or pullback. What sort of results do you get?

Also, what sort of results do you get when adding a y component? I’ve only modeled a fixed addition to the y component but in a horizontal casting plane it makes no difference. I added an optional fixed y component thinking I could modify it later to add an acceleration due to gravity if the fixed.

On the other hand, even without adding a fixed component to the y velocity any change in casting angle (assuming that the magnitude of the velocity stays the same) makes a significant difference and I find that a bit strange.

[VGB]

Just change the rotation center to the middle of the circle (that was my rotation center)_and see what you get. Dr. Perkin's and I do not disagree, we just used different centers to compute the v=omega*r values.

Gordy

I asked because momentum change occurred in Dr Perkins results but not in you do not appear to get the same outcome. I look forward to seeing how the pivot arm fits into the loop and being able to try exploiting your collision hypothesis.

Vince

Hi Walter


Thank you for putting those results together, it’s a fine piece of work. I introduced Perkins calculations to the thread to look at how AM varied around the loop, are you able to extract that calculation from your FEA?


Regards

Vince

[gordonjudd]

I do get a different result when looking at shooting line or pullback. What sort of results do you get?

Walter,

Since pull back would increase the tangential velocity around the loop while decreasing the loop velocity I would expect the AM values in a moving frame would be different than the value you would get in an earth frame. The values are also slightly different for a non-circular loop shape.

Also, what sort of results do you get when adding a y component?

Are you adding that y component in an earth frame but not in the moving frame? In that case I would think the AM values would be different.

If you are adding a y component in both frames then I would expect the AM value would be the same.

On the other hand, even without adding a fixed component to the y velocity any change in casting angle (assuming that the magnitude of the velocity stays the same) makes a significant difference and I find that a bit strange.

I find that changing the tilt angle having an impact on the AM around the loop is strange as well. Were the tangential and loop velocities (measured in the tilted direction) the same as they were for a horizontal case?

Gordy

[Walter]

Thanks Vince,

It's always nice when two different methods arrive at the same solution. I did generate a graph of am value around the loop. This compares the values in the fixed frame to those in loop centric frame. The X axis is degrees where 0 is the top of the loop and 180 is the bottom of the loop.

[Walter]

Gordy,

I think we are getting off topic but I think this discussion is definitely worth continuing so I'll start a new one.

Thanks!

Walter