Derivation of balanced torque using conservation of energy 
Let mass $m_1$ be placed at a distance of $x$ from the hinge such that the the plane makes an angle theta with horizontal
Then we drop $m_2$ at a particular distance $y$ from hinge such that after some time they remain at same height
Now potential energy of $m_2$ is
$$m_2(x+y)\sin\theta g - m_2(x)\sin\theta g$$
Similarly for $m_1$
Potential energy is
$$m_1x\sin\theta g$$
Now the potential energy should be same on both sides so that system doesn't move so we have
$$m_1x\sin\theta=m_2y\sin\theta$$
Which gives
$$ m_1x=m_2y$$
Now please tell me if this is correct or not.
I know the traditional proof is more acceptable but I want to check if I've applied the laws correctly or not. One thing I also noticed is that how can we reason for the sliding of the block like here we didn't consider it I already found one mistake that the argument that both will have same potential energy is not correct I created the equation subtracting the initial and final potential energy that account for amount of work done by gravity before coming to equilibrium but I wonder why still the result is correct is it due to that work done by gravity should be same why?
 A: If both the masses after some time remains at rest ,and as all the forces here are conservative which means conservation of mechanical energy applies in this case
So you see that in this case the gain in potential energy of $m_1$, would be equal to loss in potential energy of $m_2$ because ,if it doesn't do so then there would be a net change in potential energy which would get converted into kinetic energy, but it is not so in this case .As the question says both the masses remains at rest.
so loss in potential energy of $m_2$ as you calculated it correctly,
$$m_2(x+y)\sin\theta g - m_2(x)\sin\theta g---[1]$$
gain in potential energy of $m_1$ should be -
$$m_1x\sin\theta g----[2]$$
These two quantities $[1]$ and $[2]$ should be equal as you have done, the flaw in your argument is you have said that "the potential energy should be same on both sides so that system doesn't move" which is  wrong , in most cases we can't define absolute potential energy of system , it is always  taken with respect to some reference point like the earth itself. so the thing which you are calling potential energy is actually the difference in potential energy.
