How do, lucky for you physics is right up my street
.
The good thing is that the system you describe can be simplified greatly if the positions of the weights relative to each other dont change since you can then treat the entire system as one rigid body. To find the mass properties you could use something along these lines:
struct pointWeight
{
float m_mass;
vec m_pos;
};
float totalMass = 0.0f;
vec centerOfMass = zeroVec;
for(int i = 0; i < numPointWeights; ++i)
{
totalMass = weights[i].m_mass;
centerOfMass += weights[i].m_mass * weights[i].m_pos;
}
centerOfMass /= totalMass;
You can then apply gravitational forces at this center of mass to calculate the rotation of the rigid body. How you do this exactly depends on whether your system is 2d or 3d (your picure doesn't make that clear). I recommend reading up on rigidbody motion integration and inertia tensors, there's plenty of tutorials out there which will explain this. This of course is only important if you actualy want to simulate the motion of the system. Instead you talk about finding the time to reach equilibrium but I think you're misunderstandin some physical concepts here.
Thing is, unless your system starts out in equilibrium it will never actualy reach equilibrium given just the system you've stated. For that to happen you'd need to take into consideration energy loss of the system. The rate at which you reached equilibrium would then be a function of the efficiency of the system. Also given a perfect system, the length of the center of mass vector or indeed the mass magnitude itself will have no effect on the period of the oscillations and hence no effect on the rate at which equilibrium is reached, this is simply a function of system energy efficiency. In the real world this independance on the mass properties might not hold true however due to imperfections in the system and also the mass effect on terminal velocity should air drag be taken into account. You shouldn't have to worry about these things though, you can describe the system pretty well with just some motion damping values to represent energy loss without worrying about how exactly that energy is lost.
I'm not sure exactly what it is you're trying to acheive so I cant give you any more advice than that for now. Perhaps if you were to say what exactly you'd like to see on the screen when you run your game and how you'd like these things to interact I might be able to help further. I imagine though that what you're probably after is to simulate the system but you're just going about it the wrong way. Read up on rigidbody physics and hopefully that will give you a much better idea of what to do.
Edit: Sorry my bad, the length of the com vector does actualy effect period, but not the mass magnitude. The reason being that acceleration under gravity is the same for any mass, but a large dispacement of the mass center means it has further to travel. It still wont effect the rate of equilibrium though.
