Center of Mass Calculator (2D) finds total mass, center coordinates, and distribution for up to 3 points.
Formulas Used in Center of Mass Calculator (2D)
The calculator uses the following formulas to compute the center of mass:
Total Mass:
\\[
M = \sum_{i=1}^n m_i
\\]
Center of Mass Coordinates:
\\[
x_{\text{cm}} = \frac{\sum_{i=1}^n m_i x_i}{M}, \quad y_{\text{cm}} = \frac{\sum_{i=1}^n m_i y_i}{M}
\\]
Mass Distribution Factor:
\\[
D_f = \min\left(100 \cdot \frac{\sqrt{\sum_{i=1}^n m_i \cdot ((x_i – x_{\text{cm}})^2 + (y_i – y_{\text{cm}})^2) / M}}{R_{\text{max}}}, 100\right)
\\]
Where:
- \\( M \\): Total mass (kg)
- \\( m_i \\): Mass of the \\( i \\)-th point (kg)
- \\( x_i, y_i \\): Coordinates of the \\( i \\)-th point (m)
- \\( x_{\text{cm}}, y_{\text{cm}} \\): Center of mass coordinates (m)
- \\( D_f \\): Mass distribution factor (%)
- \\( R_{\text{max}} \\): Reference maximum distance (100 m)
Example Calculations
Example 1: Two Equal Masses, Symmetric Positions
Input: Mass 1 = 10 kg, X1 = 0 m, Y1 = 0 m; Mass 2 = 10 kg, X2 = 10 m, Y2 = 0 m; Mass 3 = 0 kg
\\[
M = m_1 + m_2 = 10 + 10 = 20 \ \text{kg}
\\]
\\[
x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{M} = \frac{10 \cdot 0 + 10 \cdot 10}{20} = 5 \ \text{m}
\\]
\\[
y_{\text{cm}} = \frac{m_1 y_1 + m_2 y_2}{M} = \frac{10 \cdot 0 + 10 \cdot 0}{20} = 0 \ \text{m}
\\]
\\[
D_f = \min\left(100 \cdot \frac{\sqrt{m_1 \cdot ((x_1 – x_{\text{cm}})^2 + (y_1 – y_{\text{cm}})^2) + m_2 \cdot ((x_2 – x_{\text{cm}})^2 + (y_2 – y_{\text{cm}})^2) / M}}{R_{\text{max}}}, 100\right)
\\]
\\[
= \min\left(100 \cdot \frac{\sqrt{10 \cdot ((0 – 5)^2 + (0 – 0)^2) + 10 \cdot ((10 – 5)^2 + (0 – 0)^2) / 20}}{100}, 100\right) = 35.36 \ \%
\\]
Result: Total Mass: 20 kg, Center of Mass: (5, 0) m, Mass Distribution Factor: 35.36%
Example 2: Three Unequal Masses, Triangular Arrangement
Input: Mass 1 = 10 kg, X1 = 0 m, Y1 = 0 m; Mass 2 = 20 kg, X2 = 10 m, Y2 = 0 m; Mass 3 = 30 kg, X3 = 0 m, Y3 = 10 m
\\[
M = m_1 + m_2 + m_3 = 10 + 20 + 30 = 60 \ \text{kg}
\\]
\\[
x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{M} = \frac{10 \cdot 0 + 20 \cdot 10 + 30 \cdot 0}{60} = 3.33 \ \text{m}
\\]
\\[
y_{\text{cm}} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{M} = \frac{10 \cdot 0 + 20 \cdot 0 + 30 \cdot 10}{60} = 5 \ \text{m}
\\]
\\[
D_f = \min\left(100 \cdot \frac{\sqrt{m_1 \cdot ((x_1 – x_{\text{cm}})^2 + (y_1 – y_{\text{cm}})^2) + m_2 \cdot ((x_2 – x_{\text{cm}})^2 + (y_2 – y_{\text{cm}})^2) + m_3 \cdot ((x_3 – x_{\text{cm}})^2 + (y_3 – y_{\text{cm}})^2) / M}}{R_{\text{max}}}, 100\right)
\\]
\\[
= \min\left(100 \cdot \frac{\sqrt{10 \cdot ((0 – 3.33)^2 + (0 – 5)^2) + 20 \cdot ((10 – 3.33)^2 + (0 – 5)^2) + 30 \cdot ((0 – 3.33)^2 + (10 – 5)^2) / 60}}{100}, 100\right) = 58.31 \ \%
\\]
Result: Total Mass: 60 kg, Center of Mass: (3.33, 5) m, Mass Distribution Factor: 58.31%
Example 3: Single Mass (Degenerate Case)
Input: Mass 1 = 50 kg, X1 = 5 m, Y1 = 5 m; Mass 2 = 0 kg; Mass 3 = 0 kg
\\[
M = m_1 = 50 \ \text{kg}
\\]
\\[
x_{\text{cm}} = \frac{m_1 x_1}{M} = \frac{50 \cdot 5}{50} = 5 \ \text{m}
\\]
\\[
y_{\text{cm}} = \frac{m_1 y_1}{M} = \frac{50 \cdot 5}{50} = 5 \ \text{m}
\\]
\\[
D_f = \min\left(100 \cdot \frac{\sqrt{m_1 \cdot ((x_1 – x_{\text{cm}})^2 + (y_1 – y_{\text{cm}})^2) / M}}{R_{\text{max}}}, 100\right) = \min\left(100 \cdot \frac{\sqrt{50 \cdot ((5 – 5)^2 + (5 – 5)^2) / 50}}{100}, 100\right) = 0 \ \%
\\]
Result: Total Mass: 50 kg, Center of Mass: (5, 5) m, Mass Distribution Factor: 0%
Related Calculators
- Tidal Force Calculator
- Orbital Velocity Calculator
- Escape Velocity Calculator
- Kinematic Equation Solver
- Physics Calculators
