Sliding weights

[Ian]

It is some years since I addressed problems in physics, but the mechanics of this problem seem both simple and complex. We are dealing with a rigid bar traveling through an arc, which approximates the segment of a circle having the shoulder at the center of that circle. The radius of that circle is the length of the arm plus the length of the blade (or, more precisely, the distance to the point of impact along the blade).

The energy from the blow will depend on the angular momentum at the point of impact. So far, so good.

If this were a problem with a ball at the end of a weightless string it would be easy to solve (think of a yo-yo or a ball flail). But we have a bar with mass along its length, and we want to add a variable mass distribution to that situation. One way to address this variation in weight distribution is to consider several scenarios, with the two extreme cases being the variable weight at each end of its travel. I'm not sure how to address the question of mass distribution along a bar. Perhaps this requires an analysis of the moments around a fulcrum (which is the point of impact), although the "fulcrum" in this case is not an immovable object but yields with the blow.

I'm sure all of these issues have been worked out previously but it is a matter of finding a reference to the solution. Presently I'm traveling but will be back in the office next week and will talk with some engineering colleagues who have far more skills in mechanics than I do. The final solution may well include calculus, so be warned.

Ian.