Energy conservation of metallic balls

[A13235378]

Homework Statement

Three small metallic balls, loaded with q charges, have masses equal to m, 4m and m. The balls are connected by insulating wires of each length and placed on a horizontal table without friction. Initially the balls are at rest in a straight line as shown. Then a quick horizontal push gives the central ball a velocity v directed perpendicular to the strings. Find the subsequent minimum distance D between the balls of dough m.

Relevant Equations

Q=mv (amount of moment)
E= Qq/4πex (potential energy)

I did energy conservation,
considering that the final velocity of the largest mass would be zero and I used moment conservation. But
I am not finding the answer . Where I maked a mistake?

 

[anuttarasammyak]

Go into COM system where conservation of momentum is satisfied. There formula of energy conservation and angular momentum conservation if you need in addition will tell you of it.

 

[etotheipi]

Your initial energy is correct, but you do not have the correct idea of what happens next. The middle ball does not necessarily come to rest at the critical point!

Here is a "shortcut" way; transform into the COM frame and write down the initial total energy and angular momentum in this frame (initial angular momentum is zero!). These will be conserved quantities.

Now consider the time where the separation of the two smaller balls are least. The relative velocities of all components will be zero, so for an instant the configuration will behave like a rigid body rotating about its centre of mass. Except the angular momentum is zero so the total kinetic energy will be $T = \frac{L^2}{2I} = 0$. Hence you can say$$E = E_{elec}(D)$$where $E_{elec}(D)$ is the electrical energy in terms of $D$ at the final configuration!

 

[TSny]

@A13235378
As @etotheipi has pointed out, the middle ball does not come to rest when the other two balls are at minimum separatation. But there is a simple relation between the speeds of the three balls at this instant. Once you see what it is, you will just need to fix up your momentum equation. Everything else looks good. Or, you can try some of the other suggestions.