Post collision angular and linear and velocities of a rotating sphere

[mattkg0]

Hi guys,

I've been looking on the internet all day for a satisfactory solution to this 2d problem but haven't found anything that makes sense.

Let's say I have a sphere.

In fact let's call it a ball with a mass m and it hits the ground with a velocity vy0 and vx0, the ball also has an angular velocity of w0 when it hits the ground.

Let's say the frictional coefficient u or horizontal coefficient of restitution ex for the ground is known.

What will be the post collision vx1 and angular w1 velocities? How are these determined? Does the change in angular velocity somehow effect the linear velocity by transferring kinetic energy through the ground?

This problem is driving me nuts the closest thing I have found is a topic in my book which works though the problem without an initial angular velocity and this link about Garwins model.

http://web.kellegous.com/ecrits/000858"

Any help on this would be greatly appreciated.

Thanks,
Matthew.

 

[AJ Bentley]

The situation is too 'messy' for an exact analytical calculation.

The precise nature of the contact with the ground, velocities, mass, elastic behaviour, friction forces will all interact in a very complicated way.

Problems of this sort in practical engineering are usually solved by experiment with particular configurations of interest. That's why we still crash-test real cars instead of relying on some guy with a calculator.

In general terms you can say that some fraction of the angular momentum will be converted into linear motion. You could even add that a high coefficient of friction would produce a greater conversion.
I don't doubt that you could eventually come up with a sophisticated theoretical model that 'worked' under ideal circumstances - but I wouldn't trust my life to it. Or even a two dollar bet.

 

[mattkg0]

Hi AJ,

Thanks for your reply and the heads-up that this is not a trivial problem to solve.

I can understand that modeling the deformation of a car as crashes into a wall is something that can't be done analytically with a calculator but I just want to model a rotating round body hitting an immovable object at a particular angle. I'm modeling the trajectory under a gravity only model, I just want something that looks and feels right and I'm having a difficult time believing that this is such an immensely difficult physics problem. It's not that hard to work out if the ball isn't rotating before the collision, why does an initial rotation introduce such complexity to the problem?

Anyway thanks again for taking the time out to reply to my post. I guess I'll keep looking and if I don't turn up anything useful I will either just implement the Garwin model above or 'make something up'.

Regards,
Matthew.

 

[AJ Bentley]

If you just want to create a simulation that 'looks right' - for a computer game for example,
there's no need to get complicated.

Depending on the exact conditions of the contact, the momentum transfer between rotation and translation will be anywhere between zero (totally frictionless contact) and some maximum where the final result is pure rolling motion.

I haven't thought about it in detail but my gut feeling is that the momentum would be shared equally between rotation and translation in that case. There's a pdf document here that might help clarify.
http://www.electron.rmutphysics.com...11 - Rolling Motion and Angular Momentum.pdf"

The vertical momentum isn't significantly affected so you can simply apply the normal rules of bouncing (coefficient of restitution etc.)

 

[mattkg0]

Hi AJ,

Thanks for that link. I think it will come in very useful especially for implementing the rolling motion of the ball after it has finished bouncing.

The bouncing of a spinning ball on the ground is such a common occurrence that I assumed there would be an established analytical model for it. However unfortunately it turns out this is not the case and even the most simplest of physical interactions in real life can turn out not match what can be mathematically predicted. From the ground I have covered in the past 3 days I think it will be a while before I take this for granted.

Fortunately for me though what I'm doing doesn't need to be 100% physically accurate (as much as I would like it to be). I'm going to implement the Garwin model which seems simple enough and should do the job, I think I understand the physics of the collision enough to 'invent' another model along the lines of what you suggested where linear and angular momentum is transferred in some way during the collision and controlled by another 'horizontal co-efficient'. I'll see what looks the best and gives me the bast control over the simulation and that will be the one I go for.

Thanks again for your help, I'm pleased I posted to this forum otherwise I would still be ripping my hair out looking for a formula and a physics book with 'the missing chapter'.

Regards,
Matthew.

 

[AJ Bentley]

even the most simple of physical interactions in real life can turn out not match what can be mathematically predicted.

Yes, that's very true.

Physics as it's taught makes it look as though modelling the real world is easy, but the truth is that the 'experiments' given in textbooks are actually carefully devised artificial situations that isolate the effect of interest so that it can be observed in a pure form.

The real world is very different. Simple interactions between ordinary phenomena, which would be quite easily resolved on their own, rapidly become confused and even chaotic.