Impact of 2 rigid beam without energy loss has multiple slns

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trytodoit
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Today, I discussed my friend about two rigid beams impact and assuming no energy loss in the impaction.
2wmhq3l.png

As in the above figure, the upper beam move down with a uniform velocity ##v_c##, and hit a beam with ##0## velocity. After that the above beam will rotate and move away, which can be described by left end velocity ##v_l## and right hand velocity ##v_r##; the lower beam will rotate around the pined point at a angular velocity ##\omega##. Therefore, we have three free variables, but only to equations, one is conservation of angular moment, the other is conservation of energy. Consequently, there are infinite combination of these three variables, which is not intuitive.

For one rigid ball hit the beam, if there is no energy loss in the impaction, there will be only one solution. Why for two beams case, there are infinite number of combinations? If there are infinite combinations, what is the property that determine which combination of the three variable for the impaction?
 
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It is probably a reasonable assumption that no horizontal force acts on the pivot during the collision (there is no friction!). That gives a third constraint, but I guess that does not help as your degrees of freedom don't consider horizontal motion at all.

Hmm... in general it is not surprising that systems are underconstrained if you have a contact line instead of a single point. Your system will react completely different if the left side would be a tiny bit ahead compared to the right side a tiny bit ahead, for example.
 
mfb said:
It is probably a reasonable assumption that no horizontal force acts on the pivot during the collision (there is no friction!). That gives a third constraint, but I guess that does not help as your degrees of freedom don't consider horizontal motion at all.

Hmm... in general it is not surprising that systems are underconstrained if you have a contact line instead of a single point. Your system will react completely different if the left side would be a tiny bit ahead compared to the right side a tiny bit ahead, for example.
The no horizontal force seems auto satisfied, which cannot introduce a equation into the system.