Violation of conservation law?

[Danyon]

Consider two objects, "A" and "B", each having mass 1kg. Object "A" moves downward towards object "B" and collides with the extended arm of object "B". Let's say for the sake of argument that during this collision object "A" lost one unit of momentum and 0.5 joules to object "B" (This value does not matter because the thought experiment works for all possible values). If conservation of linear momentum applies then object "B" should then begin to move downwards with momentum 1 and energy 0.5, the movement downwards has used all the available energy (0.5 joules) However, because object "A" struck object "B" off center, object "B" should begin rotating, this should give the object an additional rotational energy, violating the law of conservation of energy. If energy is conserved in this system then the energy of the rotation must be stolen from the energy associated with the downward movement, which would violate conservation of linear momentum. Any thoughts?

 

[phinds]

What do YOU think?

 

[Danyon]

What do YOU think?

I can't find any clean ways to resolve the problem, I've been thinking about this for months, I came across this example when I was trying to design a reactionless thruster. I hope that linear momentum is violated, it would be quite useful for spacecraft

 

[Nugatory]

Work through the algebra (and don't forget to include conservation of angular momentum). You will find that the $v^2$ term in the kinetic energy interacts nicely with the $v$ term in the momentum so that both quantities end up being conserved.

(This is, BTW, a kind of fun problem, one of those problems that makes you pause for a moment and say "Hey - that really is cool!" when you see how the math works out).

 

[Danyon]

Work through the algebra (and don't forget to include conservation of angular momentum). You will find that the $v^2$ term in the kinetic energy interacts nicely with the $v$ term in the momentum so that both quantities end up being conserved.

(This is, BTW, a kind of fun problem, one of those problems that makes you pause for a moment and say "Hey - that really is cool!" when you see how the math works out).

That's really bizarre, could you please show the math here? Just looking at the scenario its hard to imagine how something isn't violated here

 

[Nugatory]

That's really bizarre, could you please show the math here? Just looking at the scenario its hard to imagine how something isn't violated here

You're really going to have to work through the math for yourself - it's the only way to learn, which is why the Physics Forums rules prohibit giving complete solutions. But I'll push the rules a bit and give you the starting point.

Take the masses of the two objects to be $m_A$ and $m_B$. Let the initial velocity of object A to be $v_0$ and the final velocity of the two objects to be $v_A$ and $v_B$ (because object B is rotating, this will be the speed of its center of mass). Let the distance from the center of mass of object B to the impact point be $R$, the moment of inertia of object B be $I$, and the final rotational velocity of object B be $\omega$.

Now we have:
$m_Av_0=m_Av_A+m_Bv_B$ by conservation of linear momentum
$m_Av_0^2/2=m_Av_A^2/2+m_Bv_B^2/2+I\omega^2/2$ by conservation of energy
$Rm_Av_0=I\omega+Rm_Av_A$ by conservation of angular momentum.

That's three equations in three unknowns ($v_A$, $v_B$, and $\omega$)... Solve away! I guarantee you that it is worth the effort and when you succeed you'll be glad you did.

For now, this thread is closed. If you need more help from here, you should start a thread in the homework forum where the helpers can work you through the hard spots.

[The thread is closed, but I have been known to mistype formulas from time to time. Everyone watching, PM me if this is one of those times].
[EDIT - yep, I got the post collision angular momentum wrong the first time around. I just edited this post to fix it]