How to account for linear momentum in a collision?

[RDM70]

TL;DR Summary

Any object in motion has angular momentum relative to a fixed axis does it also necessarily have linear momentum.

Suppose a bar is fixed to an axle at one one so that it can pivot. The bar is initially motionless, but is set rotating about it's axle when impacted by a ball. (The ball does not strike the bar at it's pivot point.) Suppose the collision is such that the bar is set rotating and the ball is motionless after the moment of impact. (A collision with the ball traveling perpendicular to the long axis of the bar and transferring all of it's energy to the bar at the moment of impact.)

Before the impact, the ball had angular momentum with respect to the bar's pivot point. If all of that angular momentum is transferred to the bar, and the ball's motion is stopped. The ball then has no angular momentum after the impact.

Question: Before the impact, the ball was traveling in a straight line with no forces acting on it. It had angular momentum with respect to some arbitrary axis. Did it also have linear momentum? If so, what happened to the ball's linear momentum?

 

[PeroK]

Question: Before the impact, the ball was traveling in a straight line with no forces acting on it. It had angular momentum with respect to some arbitrary axis. Did it also have linear momentum? If so, what happened to the ball's linear momentum?

The bar also has linear momentum. If you think of the bar as a large number of particles, then all the particles are moving instantaneously in the same direction.

 

[jbriggs444]

Question: Before the impact, the ball was traveling in a straight line with no forces acting on it. It had angular momentum with respect to some arbitrary axis. Did it also have linear momentum? If so, what happened to the ball's linear momentum?

If the bar is mounted on a hinge or axle, then the hinge or axle may have delivered a momentary impulse to the ball+bar system as a result of the collision event.

If the bar is not mounted on a hinge or axle then, as @PeroK points out, the bar will have picked up linear momentum.

 

[PeroK]

If the bar is not mounted on a hinge or axle then, as @PeroK points out, the bar will have picked up linear momentum.

It will have linear momentum in any case.

 

[jbriggs444]

It will have linear momentum in any case.

Usually, yes. But not if the hinge/axle is positioned at the bar's center of mass.

 

[PeroK]

Usually, yes. But not if the hinge/axle is positioned at the bar's center of mass.

That's a special case where the linear momentum is zero!

The OP's scenario has the bar hinged at one end:

Suppose a bar is fixed to an axle at one one so that it can pivot.

 

[kuruman]

It will have linear momentum in any case.

But isn't there an external force acting on the bar at the pivot? In what sense does the bar have linear momentum if it's (note correct usage) rotating about the pivot?

 

[jbriggs444]

But isn't there an external force acting on the bar at the pivot? In what sense does the bar have linear momentum if it's (note correct usage) rotating about the pivot?

Its center of mass is moving.

One may choose to describe the motion of the bar as a pure rotation (no translation) about the pivot. But that description does nothing to cancel the linear momentum which exists in the ground frame regardless of what description is used.

 

[PeroK]

But isn't there an external force acting on the bar at the pivot? In what sense does the bar have linear momentum if it's (note correct usage) rotating about the pivot?

In the sense that if it came loose from its pivot Newton's first law would apply.

 

[kuruman]

In the sense that if it came loose from its pivot Newton's first law would apply.

OK.

 

[RDM70]

Thank you all for the helpful discussion!
I think I have it now.
I had not considered the impulse that might be delivered by the pivot, nor the instantaneous transitions of the bar's particles.