⚖️ Center of Mass Calculator
Calculate the center of mass for a system of point masses
Point Mass 1
How to Use This Calculator
Enter Point Masses
For each point mass, enter its X coordinate, Y coordinate, and mass value. Click "+ Add Another Point Mass" to add more points if needed.
Click Calculate
Press the "Calculate Center of Mass" button to compute the weighted average position of all point masses.
Review Results
View the center of mass coordinates (X, Y) and the total mass of the system.
Formula
X_cm = Σ(mᵢ × xᵢ) / Σmᵢ
Y_cm = Σ(mᵢ × yᵢ) / Σmᵢ
Where:
- X_cm, Y_cm = Coordinates of the center of mass
- mᵢ = Mass of the i-th point
- xᵢ, yᵢ = Coordinates of the i-th point
- Σ = Summation over all points
Example: Calculate center of mass for two points:
Point 1: (2, 3) with mass 5
Point 2: (6, 1) with mass 3
X_cm = (5×2 + 3×6) / (5+3) = (10 + 18) / 8 = 28/8 = 3.5
Y_cm = (5×3 + 3×1) / (5+3) = (15 + 3) / 8 = 18/8 = 2.25
Center of Mass = (3.5, 2.25)
About Center of Mass Calculator
The Center of Mass Calculator finds the center of mass (also called the center of gravity) for a system of point masses in two dimensions. The center of mass is the point where the weighted relative position of the distributed mass sums to zero.
When to Use This Calculator
- Physics: Calculate the center of mass for systems of particles or objects
- Engineering: Determine balance points for structures and machinery
- Mechanics: Analyze stability and equilibrium of systems
- Astronomy: Find the barycenter of planetary systems
- Robotics: Calculate balance points for robot design
- Education: Learn and practice center of mass calculations
Why Use Our Calculator?
- ✅ Multiple Points: Handle any number of point masses
- ✅ Instant Results: Get center of mass coordinates immediately
- ✅ Step-by-Step Display: See the calculation formula with your values
- ✅ Flexible Input: Add or remove point masses as needed
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Center of Mass
The center of mass is the average position of all the mass in a system, weighted by the mass at each position. It's the point where the system would balance if supported there.
- For symmetric objects with uniform density, center of mass is at the geometric center
- For systems of point masses, it's the weighted average of positions
- Heavier masses have more "pull" on the center of mass location
- In uniform gravitational fields, center of mass = center of gravity
- Critical for understanding rotation, stability, and balance
Real-World Applications
Balancing Objects: When balancing a seesaw, the center of mass must be at the pivot point. If person A (mass 60 kg) sits 2 m from center and person B (mass 80 kg) sits x m from center, the center of mass location determines balance.
Vehicle Stability: In vehicle design, the center of mass height affects stability. Lower center of mass = better stability and less rollover risk.
Projectile Motion: For rotating objects, the center of mass follows a parabolic trajectory, while the object rotates around it.
Frequently Asked Questions
What is the center of mass?
The center of mass is the point where the weighted relative position of all distributed mass sums to zero. It's the average position of all mass in a system, weighted by the mass at each position.
What's the difference between center of mass and center of gravity?
In a uniform gravitational field (like on Earth's surface), center of mass and center of gravity are the same. In non-uniform fields (like in space), they can differ. Center of mass depends only on mass distribution, while center of gravity also depends on gravitational field strength.
How does mass affect the center of mass?
Heavier masses have more "weight" in the calculation. The center of mass is pulled toward heavier objects. For example, if one mass is much larger, the center of mass will be closer to that mass.
Can the center of mass be outside the object?
Yes! For some shapes (like a donut or horseshoe), the center of mass can be outside the physical object. This happens when the shape has a hole or concave region.
What if all masses are equal?
If all masses are equal, the center of mass is simply the average of all positions: X_cm = (Σxᵢ)/n and Y_cm = (Σyᵢ)/n, where n is the number of points. This is the geometric centroid.
How do I extend this to 3D?
For 3D, add a Z coordinate and calculate: Z_cm = Σ(mᵢ × zᵢ) / Σmᵢ. The formula extends naturally to three dimensions.
Why is center of mass important?
Center of mass is crucial for understanding balance, stability, rotation, and motion. Objects rotate around their center of mass, and the center of mass follows the path determined by external forces, regardless of internal rotation.