**Centre of Mass of Three Particles at the Vertices of an Equilateral Triangle**

To find the centre of mass of three particles at the vertices of an equilateral triangle, we need to consider the masses and positions of the particles.

Given:
- Mass of particle 1 = 100 gm
- Mass of particle 2 = 150 gm
- Mass of particle 3 = 200 gm
- Side length of the equilateral triangle = 0.5 m

**Step 1: Finding the Position of Each Particle**

To find the position of each particle, we need to determine the coordinates of the vertices of the equilateral triangle. Let's assume the equilateral triangle lies in the xy-plane.

Let A, B, and C be the vertices of the equilateral triangle.

The coordinates of A can be taken as (0, 0) since it is the origin.

To find the coordinates of B and C, we need to consider the side length of the triangle. Let's assume the triangle is oriented such that side AB lies along the x-axis and side AC makes an angle of 60 degrees with the x-axis.

The coordinates of B can be taken as (0.5, 0) since it lies on the positive x-axis.

To find the coordinates of C, we can use trigonometry. The angle between the x-axis and the line segment AC is 60 degrees. The length of AC can be found using the side length of the triangle.

Using trigonometry, we can determine that the coordinates of C are (0.25, 0.25*sqrt(3)).

**Step 2: Finding the Centre of Mass**

The centre of mass of a system of particles can be found by taking the weighted average of the positions of the particles, where the weights are the masses of the particles.

Let (x1, y1), (x2, y2), and (x3, y3) be the coordinates of particles 1, 2, and 3, respectively.

The x-coordinate of the centre of mass, Xm, can be calculated using the formula:

Xm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)

Similarly, the y-coordinate of the centre of mass, Ym, can be calculated using the formula:

Ym = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3)

Substituting the values, we get:

Xm = (100*0 + 150*0.5 + 200*0.25) / (100 + 150 + 200)

Ym = (100*0 + 150*0 + 200*0.25*sqrt(3)) / (100 + 150 + 200)

Simplifying the equations, we find:

Xm = 0.3 m
Ym = 0.0866 m

Therefore, the centre of mass of the system of particles is located at (0.3 m, 0.0866 m).

**In Summary**

The centre of mass of the system of particles located at the vertices of an equilateral triangle with side length 0.5 m and masses 100 gm, 150 gm, and 200 gm is found to be at coordinates (0.3

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