Question

Three point masses M1, M2, and M3 are located at the vertices of an equilateral triangle of length a. Then what will be the coordinates of the centre of mass in this system?

Answer 1

The center of mass of three point masses at the vertices of an equilateral triangle of length a will be at the coordinates (a/2, sqrt(3)a/6), which is the centroid of the triangle.

Step-by-step explanation:

To determine the coordinates of the center of mass of three point masses located at the vertices of an equilateral triangle, one can use the properties of symmetry of the equilateral triangle and the definition of the center of mass for a system of particles. Given that all three point masses, M1, M2, M3, are equidistant from each other and let's say they're at the vertices of the equilateral triangle, we use the following steps:

• Choose a convenient coordinate system, typically with the origin at the center of the equilateral triangle for symmetry.
• Since the triangle is equilateral, by symmetry, the center of mass will be at the centroid of the triangle, regardless of the masses.
• The centroid divides each median in a 2:1 ratio, so we calculate the coordinates by finding the average of the vertices' coordinates.

If the vertices of the equilateral triangle are defined at coordinates A(0, 0), B(a, 0), and C(a/2, sqrt(3)a/2), the coordinates of the center of mass (x_cm, y_cm) can be found using:

x_cm = (x1 + x2 + x3) / 3 = (0 + a + a/2) / 3 = (3a/2) / 3 = a/2

y_cm = (y1 + y2 + y3) / 3 = (0 + 0 + sqrt(3)a/2) / 3 = (sqrt(3)a/2) / 3 = sqrt(3)a/6

Thus, the center of mass will be at (a/2, sqrt(3)a/6) in the coordinate system with the origin at the center of the triangle.