Elastic Collision: Rotational Momentum and Linear Momentum

[luckis11]

When two equal masses collide without having any rotational velocity before the collision, and they do attain some rotational velocity after the collision, then do we subtract the rotational momentum from the linear momentum which is given by the known equations, or is the rotational momentum procuded in addition to the linear momentum?

 

[cepheid]

then do we subtract the rotational momentum from the linear momentum which is given by the known equations, or is the rotational momentum procuded in addition to the linear momentum?

You can't subtract two physical quantities that don't have the same dimensions (this makes no more sense than subtracting a time from a length). I believe that the answer is that angular momentum is conserved separately. Since there was none to begin with, the angular momenta of the two masses must sum to zero.

Linear momentum, which describes the translational motion of the centre of mass of each object, is also conserved. Conservation of (linear) momentum would be applied to this problem in the normal way in order to obtain the speed and direction of translation of the centre of mass of each object.

 

[luckis11]

Is this just your opinion or an established theory of physics based on logic? If it is established, has it also been prooven experimentally?

 

[cepheid]

How about "classical mechanics?" Have you heard of Newton's second and third laws? I think that conservation of momentum follows quite readily from them. The bottom line is that if these two objects collide, then in the absence of any external forces, momentum is conserved. In the absence of any external torques, angular momentum is also conserved. Yes I'm sure it has been proven experimentally/put to practical use (flywheels and innumerable other things) countless times in the past four centuries.

 

[luckis11]

Any links clarifying this with an example?

 

[Dale]

http://hyperphysics.phy-astr.gsu.edu/Hbase/conser.html

 

[luckis11]

I just found strange that when a moving ball having a velocity u at the axis of x, hits a still ball non-metopically, then after the collision two equal momentums are born out of zero at the axis of y, that were not existent before the collision. "But this does not violate the conservation of momentum because they have the opposite direction".

And you are saying that in this case, not only these two mementums at the axis of y are born out of zero, but also angular, rotational momentums are born out of zero. Have I got it right?

 

[Majcah]

If the two objects are spherical in shape and collide in such a fashion as to result in return angles that differ from their approach, then I believe that the nature of the material can produce angular momentum due to rolling friction.

 

[luckis11]

Isn't the angular momentum of the still ball (at least also) caused by the fact that during the collision, one part of the still ball attains part of the linear velocity of the moving ball, whereas the other part of the still ball is not? Then it also seems reasonable that the angular momentum is subtracted from the linear momentum, i.e. some linear momentum became angular. Both of the two different answers seem reasonable. If I could handle all the seeming logical complications of the problem, I wouldn't ask which answer is the correct one.

Actually I only want the answer of classical physics. What happens in reality, is very doubtful for me. For example, I doubt that two opposite momentums can be born out of zero. And if (as it seems) this has indeed been prooven experimentally, it is very possible that rest momentums of the balls became linear momentums of the balls.

 

[luckis11]

When a moving ball hits a still ball of equal mass obliquely, having a velocity u before the collision at the axis of x, then the sum of the moduluses (absolute values) of the two resultant momentums of the two balls after the collision is greater than the sum of the moduluses of the resultant momentums of the two balls before the collision. Does this increase of the sum of the moduluses of momentums (comparing the before and the after the collision sum), happen only in the case that one of the two balls is still before the collision, or does it also happen when both balls are moving before the collision?

 

[cepheid]

Isn't the angular momentum of the still ball (at least also) caused by the fact that during the collision, one part of the still ball attains part of the linear velocity of the moving ball, whereas the other part of the still ball is not?

The best way to think about it is in terms of torques. Do you know what the relationship between torque and angular momentum is? If you do, you will realize that the only way for the still ball to gain some angular momentum is if there is a net torque exerted upon it. This will happen if the moving ball hits it off-centre, meaning that there is a *force* acting on the still ball a certain *distance* away from the centre of mass. This force acting at a distance produces a net torque around the centre of mass and sets the ball spinning.

Then it also seems reasonable that the angular momentum is subtracted from the linear momentum, i.e. some linear momentum became angular.

Why does this seem reasonable? You haven't really explained that. Remember what I said about how adding or subtracting physical quantities that have different dimensions is meaningless?

Both of the two different answers seem reasonable. If I could handle all the seeming logical complications of the problem, I wouldn't ask which answer is the correct one.

What two answers are you referring to? It's not clear in your post.

Actually I only want the answer of classical physics. What happens in reality, is very doubtful for me.

What? Why? If the predictions of classical physics had not been borne out by experiment, then it would have been rejected in favour of something that did explain what we observe in reality. But this did not happen, because as far as we know, there is a well-defined range of energies and length scales over which classical mechanics is perfectly applicable and describes nature with great precision. Look at the success of Newtonian gravitation in explaining the motions of the planets, for example. There is no logical reason for you to doubt that what classical mechanics says about colliding balls should be any different from "reality."

For example, I doubt that two opposite momentums can be born out of zero.

Why not? They ADD UP to zero, don't they? Remember, momentum is a VECTOR quantity.

And if (as it seems) this has indeed been prooven experimentally, it is very possible that rest momentums of the balls became linear momentums of the balls.

Here you are not using standard terminology. What do you mean by "rest momentums?" As far as I know this term is not meaningful.

 

[luckis11]

Forget about it all, it is needed to say a lot to explain what I mean. What is the answer to the question of my previous post?

 

[Doc Al]

When a moving ball hits a still ball of equal mass obliquely, having a velocity u before the collision at the axis of x, then the sum of the quantity of the two resultant momentums of the two balls after the collision is greater than u.

Are you saying that the total momentum after the collision is greater than before the collision? If so, that's incorrect.

Does the birth out of zero of two opposite component linear momentums, and the increase of the total quantity of momentum, happen only in the case that one of the two balls is still, or it also happens when both balls are moving before the collision?

Momentum is conserved; the total quantity of momentum (which is a vector) does not change.

Note that momentum is not simply mass*speed, but mass*velocity. The direction makes a big difference!

 

[Dale]

I just found strange that when a moving ball having a velocity u at the axis of x, hits a still ball non-metopically, then after the collision two equal momentums are born out of zero at the axis of y, that were not existent before the collision. "But this does not violate the conservation of momentum because they have the opposite direction".

Why should you find this strange. It happens all the time. In billiards, with rockets, in car crashes, etc. That is exactly how momentum works and it is clearly demonstrated all the time in nature.

And you are saying that in this case, not only these two mementums at the axis of y are born out of zero, but also angular, rotational momentums are born out of zero. Have I got it right?

Yes. If one object exerts a clockwise torque then the other object exerts a counter-clockwise torque. The sum of the two torques is 0 and therefore angular momentum is conserved. This follows from Newton's 3rd law.

 

[cepheid]

Okay, as requested, I will look only at this post:

When a moving ball hits a still ball of equal mass obliquely, having a velocity u before the collision at the axis of x, then the sum of the quantity of the two resultant momentums of the two balls after the collision is greater than u. The sum of the two component linear momentums of the two balls at the axis of x after the collision have the same direction and equal u. Thus two opposite and equal component linear momentums are born out of zero at the axis of y.

Did you make up this statement, or is it a quote from somewhere?

Does the birth out of zero of two opposite component linear momentums, and the increase of the total quantity of momentum, happen only in the case that one of the two balls is still, or it also happens when both balls are moving before the collision?

The total quantity of momentum does not increase. That is the whole point. In every scenario, momentum is *conserved*, and, in the case of an elastic collision, energy is conserved. These two facts are what allow you to determine what will happen.

If one ball is still, and the other ball is moving in a straight line along the x axis, then there was initially zero momentum along the y-axis before the collision. That is why the y-components of the momenta after the collision have to be equal in magnitude and opposite in direction, so that momentum will be conserved.

If both balls initially have some momentum in the y-direction, then the final y-momenta do not have to add up to zero. They just have to add up to the same value as the initial y-momenta (again, momentum is conserved).

 

[luckis11]

Ok, substitute the word "quantity" with "absolute values".

 

[Doc Al]

Ok, substitute the word "quantity" with "absolute values". The absolute value of -3 is 3. Is "absolute value" the correct name for this?

You haven't answered that last question of mine.

Please rephrase your question clearly. Don't use the term "momentum" if that's not what you mean.

 

[luckis11]

When a moving ball hits a still ball of equal mass obliquely, having a velocity u before the collision at the axis of x, then the sum of the moduluses (absolute values) of the two resultant linear momentums of the two balls after the collision is greater than the sum of the moduluses of the resultant linear momentums of the two balls before the collision. Does this increase of the sum of the moduluses of linear momentums (comparing the before and the after the collision sum), happen only in the case that one of the two balls is still before the collision, or does it also happen when both balls are moving before the collision?

 

[Dale]

The sum of the moduluses of the momenta can increase or decrease or stay the same depending on the details of the interaction, you cannot make any general statements about it. The sum of the moduluses of the momenta is not a useful quantity and is not conserved. Momentum is a vector quantity and must be treated as such.

 

[luckis11]

You did not answer the question.

Also, in which cases this sum is reduced after the collision instead of increased? I cannot find any case that it is reduced (always referring to perfectly elastic collisions).

 

[cepheid]

You did not answer the question.

Yes he did. He said, "It depends on the situation and you can't make any general statements about it."

 

[Bob S]

You haven't answered that last question of mine: Does the increase of the sum of the absolute values of momentums also happen in the case that both two balls are moving before the collision?

Do a linear transformation into a coordinate system in which the center of gravity of the system is at rest (and both balls are moving) before the collision. So the absolute value of the sum of momenta and angular momenta is zero, before, and after. If you have billiard balls, and sliding friction with the surface, only then is linear momentum (and energy) converted to angular momentum (and rotational energy).

 

[luckis11]

Ok, I'll tell you one of the main points that troubles me. If the sum of these absolute values of momentums increases in every oblique collision, then if we close a given number of circles in a box and each circle comes back from the wall of the box with the same absolute value of velocity with which it hit the wall, then the average velocity of the circles will keep increasing as the number of collisions between the circles increases. But this doesn't seem to happen in simulations of elastic collisions of circles closed in a box. How is this possible?

 

[Dale]

Because your assumption is incorrect. I already told you that it can increase, decrease, or stay the same and that you cannot make any general statements about it. Your thought experiment should have convinced you of that.

Really, the sum of the absolute value of the momentum is a completely useless concept. You are just confusing yourself and any other student reading this thread for absolutely no purpose.

 

[luckis11]

I asked you in which case the sum will decrease, and you did not answer. I ask again: IN WHICH CASE?

 

[Dale]

The way I would approach this if I were inclined would be as follows:
I would write the equations for the collision in the center of momentum frame. The inputs would be:
3 components of initial velocity for mass A
2 masses for particles A and B
2 scattering angles
These 7 parameters are sufficient to specify a perfectly elastic collision in the center of momentum frame. Then add 3 components of boost to obtain the result in the lab frame. That is a total of 10 parameters to specify the problem. Without loss of generality you can normalize the mass and velocity of A in the center of mass frame reducing the number to 8. You can also specify the initial velocity of A to be in the positive x direction further reducing the number to 6 without loss of generality. I would calculate the change in the sum of the magnitudes of the momenta as a function of those 8 parameters and minimize it.

If you are really interested feel free to do this calculation. You should find about half of the 6 dimensional parameter space that leads to reducing the sum and half that leads to an increase.

 

[luckis11]

I wish I could, but I cannot understand you. Can't you discribe with words the solution of the paradox?

I cannot see any solution. Suppose there are only two circles in the box. In which case the sum will decrease after a collision between them?

 

[Dale]

I thought of an easier way than the way I proposed above. Take any example where the sum increases. Define your new initial state as the negative of the original final state and define your new final state as the negative of the original initial state. You now have a new collision where momentum and KE are still conserved, but the sum decreases.

 

[Doc Al]

Good thinking, DaleSpam. Genius!

 

[Dale]

DaleSpam. Genius!

I do like the sound of that!