Center of Mass Calculator
Calculate the center of mass for discrete masses in 1D, 2D, or 3D space
Center of Mass Calculator
Choose coordinate system
Maximum 10 masses
Mass / Distance units
Mass Properties
Center of Mass Results
✅ Equal masses detected! Center of mass = Centroid
Centroid: (0.5000, 0.5000) m
Formula used:
• X_com = (Σ m_i × x_i) / M = 0.5000 m
• Y_com = (Σ m_i × y_i) / M = 0.5000 m
• Total Mass (M) = Σ m_i = 2.000 kg
Example Calculation
Two Objects on a Seesaw
Problem: Find the center of mass of two children on a seesaw
Child 1: 30 kg at position x = 0 m
Child 2: 20 kg at position x = 4 m
Solution
1. Total mass: M = 30 + 20 = 50 kg
2. Center of mass: X_com = (30×0 + 20×4) / 50
3. X_com = (0 + 80) / 50 = 1.6 m
Result: Center of mass is 1.6 m from the first child
Key Concepts
Center of Mass
Point where all mass can be considered concentrated
Centroid
Geometric center for objects with uniform density
Balance Point
Where object balances perfectly
Formula Reference
General Formula
COM = (Σ m_i × r_i) / M
1D Formula
x_com = (Σ m_i × x_i) / M
Centroid (Equal Masses)
Centroid = Σ r_i / N
Applications
Designing playground equipment (seesaws, swings)
Vehicle stability and handling analysis
Structural engineering and crane operations
Aerospace and rocket design
DIY projects and mobile design
Understanding Center of Mass
What is Center of Mass?
The center of mass is the unique point at which the weighted relative position of the distributed mass sums to zero. This is the point where the entire mass of a body or system could be considered to be concentrated. When a force is applied to this point, it produces only linear acceleration with no rotation.
Key Properties
- •Represents the average position of mass distribution
- •Closer to heavier masses in the system
- •For uniform density objects, equals the centroid
- •Independent of coordinate system choice
Mathematical Foundation
General Formula
r_com = (1/M) × Σ m_i × r_i
Component Form
x_com = (Σ m_i × x_i) / M
y_com = (Σ m_i × y_i) / M
z_com = (Σ m_i × z_i) / M
Center of Mass vs. Centroid
- Center of Mass: Considers mass distribution
- Centroid: Pure geometric center
- When Equal: Uniform density objects
- Physics Relevance: COM determines motion
Real-World Applications
🏗️ Engineering
- • Structural stability analysis
- • Bridge and building design
- • Crane load calculations
- • Machine balance optimization
🚗 Transportation
- • Vehicle handling characteristics
- • Aircraft balance and control
- • Ship stability calculations
- • Load distribution planning
🏃 Sports & Recreation
- • Athletic performance analysis
- • Equipment design optimization
- • Playground safety calculations
- • Balance training applications