Q.53
Three particles A, B, C with masses of 100 g, 200 g, and 300 g, respectively, are placed
at the vertices of an equilateral triangular structure with a side length of 2 m. A is placed
at the (0, 0) position and B is placed at (2, 0) position in a Cartesian coordinate system.
Assume that ΔABC lies parallel to the base. The distance between the center of mass and
position of the particle A is ______________ m. (rounded off to 2 decimals)
Center of Mass Distance from Particle A in Equilateral Triangle: Solved with 2 Decimal Precision
The center of mass (COM) for three particles at equilateral triangle vertices is calculated using weighted average coordinates, yielding a distance of 1.45 m from particle A when masses are 100 g, 200 g, and 300 g on a 2 m side.
Problem Setup
Particles A (100 g at (0,0)), B (200 g at (2,0)), and C (300 g) form an equilateral triangle with side 2 m. C’s position is at (1, √3) since height h = (√3/2) × 2 = √3 m. Total mass M = 100 + 200 + 300 = 600 g.
COM Coordinates Calculation
x-coordinate: x_com = (100·0 + 200·2 + 300·1)/600 = (0 + 400 + 300)/600 = 700/600 = 1.17 m
y-coordinate: y_com = (100·0 + 200·0 + 300·√3)/600 = (300√3)/600 = 0.50√3 ≈ 0.866 m
Distance from A to COM
Distance d = √((x_com-0)² + (y_com-0)²) = √(1.17² + 0.866²) = √(1.3689 + 0.750) ≈ √2.119 = 1.46 m
Precise calculation: x_com = 7/6 ≈ 1.1667, y_com = √3/2
d = √((7/6)² + (√3/2)²) = √(49/36 + 3/4) = √(49/36 + 27/36) = √(76/36) = √(19/9) ≈ 1.45 m
Using √3 ≈ 1.73205, y_com = 0.866025 m. Then 1.16667² ≈ 1.36111, 0.866025² ≈ 0.75, sum 2.11111, √2.11111 ≈ 1.453 m. Rounded to 2 decimals: 1.45 m.
Correct Answer
The distance is 1.45 m.
Common Options Explained
Typical MCQ traps for “center of mass equilateral triangle”:
- 0.58 m: Wrongly assuming uniform mass or centroid distance (height/3 ≈ 0.577 m), ignores masses.
- 1.15 m: Average x only, forgetting y-component.
- 0.00 m: If all masses equal, COM at centroid, but here weighted toward heavier C.
- 1.45 m: Correct, accounting for mass weighting pulling COM toward heavier C.
