Two equal-mass particles, each of mass m, occupy the vertices at the base of an equilateral triangle. The triangle's center of mass is located midway between the base and the third vertex. What is the mass at the third vertex?

Let's assume that the equilateral triangle has side lengths of "a," and we have two equal masses, m, at two of the vertices on the base. We are trying to find the mass at the third vertex, let's call this mass "M."

For an equilateral triangle, the center of mass (COM) is located at the centroid, which, due to the symmetry, is at the same distance from all three vertices. The centroid divides each median (line from a vertex to the midpoint of the opposite side) in the ratio 2:1. Since the COM is said to be midway between the base and the third vertex, this implies that the mass at the third vertex alone is balancing the masses at the base.

Let's assign coordinates to the vertices for easier visualization. Let the two base vertices with mass m be at coordinates (0,0) and (a,0), and the third vertex with mass M be at (a/2, √3/2 * a). The COM's y-coordinate can then be calculated using the centroid formula for the y-coordinate.

Y coordinate of COM = (m * 0 + m * 0 + M * √3/2 * a) / (m + m + M)

Since the COM is located midway along the median, its y-coordinate must be √3/6 * a, which is half the vertical height of the triangle.

√3/6 * a = (M * √3/2 * a) / (2m + M)

By simplifying the equation and cancelling out the common factors, we get:

1/3 = M / (2m + M)

Multiplying through by the denominator, we have:

2m + M = 3M

This simplifies to:

2m = 2M

Dividing both sides by 2:

m = M

Therefore, the mass at the third vertex, M, must be equal to each of the masses at the base, which is m.

Extra: In classical mechanics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point at which the weighted relative position of the distributed mass sums to zero. When analyzing systems like the one in this problem, we can use the center of mass concept to determine how masses are distributed in relation to that point, particularly in symmetric configurations such as triangles.

In the case of an equilateral triangle, the symmetry allows us to conclude that if the center of mass for two particles lies on the line that is also the median from the third particle, and it's given that the center of mass is equidistant from the two masses at the base and the third vertex, then the masses must be equal. This is because in a uniform gravitational field, the center of mass is the point that moves as if the entire system's mass were concentrated at that point and all external forces were applied at that point.

This concept is particularly important in physics, as it simplifies the analysis of the motion of complex systems of particles by allowing us to treat the entire mass as if it were concentrated at the center of mass. It also simplifies the calculation of the gravitational force in astrophysics and the analysis of stability and balance in engineering.