Collisions With Friction and Spin

Question:

I am currently studying at Thomas Mills High School Framlingham in England. I have just started my second year of a two year physics A level. Next year I will be going to university (possibly in the USA). I am currently part way through an investigation into the collision between two billiard balls. I have built and designed a contraption to release a snooker cue to hit a billiard ball with exactly the same force each time. My task is to investigate the relationship between the force applied on the original ball and the force applied to the second ball. The variables I have chosen to look at is the angle at which the original ball strikes the second ball (so it won't be a dead centre collision), and the position the cue hits the original ball so different types of spin are exhorted on the original ball.

After reading your Physics 1 guide my understanding of collisions has become clearer as I only have a basic level of understanding. I was wondering if you would be able to suggest any ideas on how to accurately take my required measurements and to also suggest any formulas on the direction of velocity of both balls when they collide when spin is involved. I understand my explanation may not be that clear, this is because I am not 100% myself how I am going to carry out my task, If you need to know anything more to help assist you with the answering of my questions please just ask. If you could help with this I would be very grateful.

With regard to recording data from a billiard ball collision, you might try laying out a grid on the table and video taping the collision from directly overhead. The shutter speed of the video camera serves as a natural time mark so that velocities and accelerations can be determined.

In the collisions commonly covered in undergraduate physics, even though we refer to them as billiard ball collisions, the total disregard of friction allows us to ignore any spin that the incoming ball might have since it cannot play any role in the collision in the absence of friction between the balls. To account for spins we must open the whole issue of friction.

When spin is involved in a collision there is another conservation law that comes into play... the conservation of angular momentum. The transfer of this momentum requires that tangential forces act on the ball and tangential forces imply friction. Of course the presence of friction means that the simple zero-sum energy conservation law does not hold. So we lose one of our known relationships between the before collision and after collision parameters.

The cue at the instant of impact has some angular momentum relative to the center of the ball being struck if it is moving along a line that does not contain the center of the ball. Immediately after the collision the cue will have a smaller angular momentum relative to the ball center and the ball will have gained the difference as angular momentum about its own center. There will be a similar transfer of angular momentum at the collision between the cue ball and the target ball.

Now we get into a significant complication. In the presence of friction, the motion of the cue ball along the table surface will induce a spin since the ball rolls rather than sliding. This spin will have associated with it an angular momentum vector parallel to the table surface. The total angular momentum of the cue ball then is the sum of the angular momentum imparted by the cue and that picked up from the table. This vector may point in any direction depending on the point where the cue struck the cue ball. The reaction of this complex spin with the table may result in a curved path of both the cue ball and the target ball.

I have not tried to solve this problem but I suspect that it is messy in the extreme. It will be very difficult to quantitatively predict from a particular cue velocity and point of impact on the cue ball, the trajectory of the cue ball and the target ball. You might try to guess which of the complicating factors are small enough to be neglected and see how your results agree with experiment. If you decide to proceed along this path let me know. I might be able to make some guesses to guide you.

You have come upon one of the cruel realities of physics, that it is easy to come up with an experiment that requires unmanageable mathematics to explain. You might also gain some insight into what a remarkable computational engine the human brain is. Somehow good billiards players solve this intractable problem in a few seconds.