Derive an equation of center of …

The center of mass (CoM) is not an equation, it's a definition. The center of gravity, on the other hand, is an equation. So yes, that formula is just a definition, but it is not a random one. After you define it, it turns out to have multiple nice properties. But that's not by chance, but because of that special form. And that's because it is the weighed average position. That means, it is the average position, but weighed according to masses. If you have two masses, m1 and m2, in positions 1 and 2, then you would easily compute average in the same way you calculate your average score in school: r⃗ mean=r⃗ 1+r⃗ 22 And it is obviously right in the middle of them. Half the distance. Notice that this is the same as r⃗ mean=12(r⃗ 1+r⃗ 2) Now, what if m2 is twice m1? Then m2 is like having two units of m1 placed in the same site. That's like having three particles: r⃗ mean=r⃗ 1+r⃗ 2+r⃗ 23=r⃗ 1+2⋅r⃗ 23 And that's the same as r⃗ mean=13(r⃗ 1+2⋅r⃗ 2)=13r⃗ 1+23r⃗ 2 Can we generalize this? What if it is m2=1.5m1? or any other? Check that 1/3 and 2/3 are precisely m1/M and m2/M, where M is the total mass. So yes, we have r⃗ CM=m1Mr⃗ 1+m2Mr⃗ 2+⋯=1M∑imiri→ The usual formula. It's a definition, yes, but it is the average, and that makes it have special properties that other definitions would not have.