Problem with angular momentum conservation law.?

Question

Hello,

I have got a following physics problem.

A dumbbell (distance between balls is l) stands upright on a flat. This dumbbell is released to move. What speed will have each ball when the dumbbell hits the flat (there is no friction).

I thinks that here should be used energy and angular momentum conservation laws, but I have now idea how to use these equations.

Has anybody any ideas how to solve it?

Answer 1

I'll get you started and then see if I can get a more complete answer later.

Here's how I modeled the problem:
The dumbbell is two point masses each of mass m separated by a weightless rod of length L.
No friction on the flat surface so the dumbbell will fall and rotate such that the center of mass will be constant in the horizontal and the energy of the system will be conserved
So
m*g*h=.5*m*v1^2+.5*m*v2^2, where v1 is the speed of the falling mass and v2 is the speed of sliding mass.
2*g*h=v1^2+v2^2

The dynamic of the system is such that the force pushing the lower ball will be translated through the connecting rod.

The force in the rod is Cos(th)*m*g-m*w^2*L, where th is the angle the upper ball has rotated downward, and w is the instantaneous angular speed of the upper ball. Note that the angular speed of the upper ball lowers the force in the connecting rod.. The horizontal component of the force is Sin(th)*(Cos(th)*m*g-m*w^2*L).

Let's look at the end condition where th=pi/4 (90) sin(90)=1 and cos(90)=0, so the force in the rod is decelerating the lower ball. In the special case of no friction and equal mass, the force on the lower ball will cause the ball to move and then stop the instant the upper ball is at 90 degrees from the start. Therefore v2=0 and
v1=sqrt(2*g*L)

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Answer 2

I don't think you should use conservation of angular momentum since you have a net torque (gravity) acting on the dumbbell. All you need to do is say that the change in potenial energy (mgh) is equal to the change in kinetic energy (1/2mv^2). The ball already on the ground will simply have a final speed of zero...

Answer 3

3. Conservation of angular momentum can be deduced from Noether's theorem applied to the system's lagrangian, by considering a small change in the angular coordinates.