Determine the position of the center of gravity of a system of bodies consisting of two balls of masses

Determine the position of the center of gravity of a system of bodies consisting of two balls of masses m and 2m, put on the ends of a non-tangential rod of length l. The diameters of the balls are considered small in comparison with the length of the rod.

Given:
m1 = m is the mass of the ball at one end of the rod;
m2 = 2 * m is the mass of the ball at the other end of the rod;
g = 10 m / s ^ 2 – acceleration of gravity;
l is the length of the weightless rod.
It is required to determine the position of the center of gravity X.
Since, according to the condition of the problem, the rod is weightless and the diameters of the balls are negligible compared to the length of the rod, the center of gravity will be such a point, at the position of the stop on which the rod will be in equilibrium (that is, it will be a lever). Then:
F1 * x = F2 * x1, where F1, F2 are the gravity forces acting on the balls, and x and x1 are the distance to the stop point (center of gravity).
By the condition of the problem, we have that x + x1 = l, hence x = l – x1. We carry out the transformation of the equation:
F1 * x = F2 * (l – x)
F1 * x = F2 * l – F2 * x
F1 * x + F2 * x = F2 * l
x * (F1 + F2) = F2 * l
x = F2 * l / (F1 + F2) = m2 * g * l / (m1 * g + m2 * g) = 2 * m * g * l / (m * g + 2 * g * m) = 2 * m * g * l / 3 * g * m = 2/3 * l.
Answer: the center of gravity will be at a distance of 2/3 * l from a ball with a mass m or, accordingly, at a distance of 1/3 * l from a ball with a mass of 2 * m.