Rod elastically impacting a ball - impact location

[davidwinth]

TL;DR

If a cylindrical rod travels at a steady speed perpendicular to its axis, does the impact force felt by an initially stationary ball matter?

Say we have a cylindrical rod of length L and mass M traveling at constant speed S in a direction that is perpendicular to the axis. Further, say that the rod impacts an initially stationary ball of mass m, and the two collide elastically. My question is: does the force experienced by the ball depend on where on the rod the impact takes place?

My intuition says that if the ball and rod impact at, say, L/2 (in the middle), the ball would experience a greater force than if the impact was right at the end of the rod at L. This is because when the ball hits the rod in the end, some of the impact energy goes into imparting a rotation to the rod. When the ball hits in the middle, there is no imparted rotation and the rod only slows down somewhat.

I tried to work this out with math but couldn't quite figure out how to get the right variable relations. As I see it, there are three equations we can use: conservation of energy, conservation of angular momentum, and conservation of linear momentum. On the other hand, there are three unknowns: The final speed of the rod, the final speed of the ball, and the final angular velocity of the rod.

So by the COE and COLM, we can relate the initial (known) velocities to the final (unknown) velocities. That part is ok to me. But I don't understand how to relate the final angular velocity of the rod to anything. It seems to me that the COE and COLM are two equations in two unknowns and therefore determine both final velocities no matter how the rod rotates after the collision. But if the final velocity of the ball is the same no matter where the impact takes place, then the force on the ball is the same no matter where the impact takes place. If the force is the same no matter where the impact takes place, then the torque on the rod varies only by impact location since torque depends on force and distance from axis of rotation, which always passes through the rod centroid. So we have a situation where the ball always feels the same force and yet the rod will not spin at the same rate after the impact independent of impact location! That seems contradictory to me.

I must be missing something. I'd appreciate any input.

 

[haruspex]

First, let’s tidy the question up. The impact is given as elastic. The impact takes some time. The force the two bodies exert on each other varies over that time, reaching a maximum at the half way point. So maybe you are asking how the maximum force depends on impact location, but it would be cleaner to stop referring to force and consider impulse - the integral of the force over the duration of the impact - instead.
The impulse transfers momentum. For a given impact location, you have three unknowns: the speed of the ball afterwards (its direction is obvious), the speed of the mass centre of the rod afterwards (ditto) and the rate of angular rotation of the rod afterwards.
You have three equations available: conservation energy, linear momentum and angular momentum. To be safe, take some fixed point in space (I suggest either the initial position of the ball or the position of the rod's mass centre at the moment of impact). Before impact, neither object has angular momentum about that axis; after impact, it depends which axis you choose.
If you choose the rod's mass centre, both have angular momentum about it (for the ball, it's the linear momentum multiplied by the perpendicular distance to the axis), and these two angular momenta must be equal and opposite.
If you choose the ball's initial position, it will not acquire angular momentum about that point, but the rod will have angular momentum in one direction due its spin and equal and opposite angular momentum due to its linear momentum multiplied by the perpendicular distance from the momentum to the axis.

I suggest trying both to see how the two produce the same answer.

 

[wrobel]

TL;DR: If a cylindrical rod travels at a steady speed perpendicular to its axis, does the impact force felt by an initially stationary ball matter?

My question is: does the force experienced by the ball depend on where on the rod the impact takes place?

Why don't you write down formulas?