Center of Mass Calculator
Find the center of mass for point masses in 2D or 3D, or pick a standard shape like a semicircle, cone or triangle. Full step-by-step moments table included.
Center of Mass
Enter values to calculate
Formula
x_cm = Sum(m_i * x_i) / Sum(m_i)
The center of mass is the weighted average of all mass positions. Each coordinate (x, y, z) is calculated independently using the same formula.
Point Masses
Enter any number of masses with x, y (and z) coordinates. See the weighted moment for each mass and the resulting center of mass.
Standard Shapes
Pick from 8 geometric shapes including semicircle, hemisphere, cone, cylinder, triangle, rectangle and circle with known centroid formulas.
Step-by-Step Table
View individual moments (m*x, m*y, m*z), totals and the final division for a complete walkthrough of the calculation.
Center of Mass for Common Shapes
| Shape | Center of Mass Location | Formula |
|---|---|---|
| Uniform Rod | Midpoint | x = L/2 |
| Rectangle | Geometric center | x = w/2, y = h/2 |
| Triangle | Centroid (1/3 from each side) | x = (x1+x2+x3)/3 |
| Circle / Disk | Geometric center | At the center point |
| Semicircular Disk | 4r/(3pi) from diameter | y = 0.4244r |
| Semicircular Wire | 2r/pi from diameter | y = 0.6366r |
| Solid Hemisphere | 3r/8 from flat face | z = 0.375r |
| Solid Cone | h/4 from base on axis | z = 0.25h |
| Hollow Cone Shell | h/3 from base on axis | z = 0.333h |
| Solid Cylinder | Midpoint of height on axis | z = h/2 |
How to Find the Center of Mass
Understanding Center of Mass in Physics
What Is the Center of Mass?
The center of mass is the single point where you could balance an entire object (or system of objects) if you supported it from below. For a uniform square plate, it is right in the middle. For a baseball bat, it is closer to the thicker end. Every physical object has a center of mass, and it plays a central role in how things move, rotate and balance.
In physics, the center of mass simplifies complex motion. When you throw a wrench through the air, it spins and tumbles, but its center of mass follows a smooth parabolic path, just like a ball would. All the complicated rotation happens around that one special point.
The Formula for Discrete Point Masses
For a system of n point masses, the center of mass coordinates are:
x_cm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn)
The same formula applies independently to y and z. Each coordinate of the center of mass is the mass-weighted average of that coordinate across all objects. Heavier objects pull the center of mass closer to their position.
Why It Matters
The center of mass shows up across physics and engineering:
- Mechanics: Newton's second law (F = ma) applies to the center of mass of a system. External forces accelerate the center of mass, regardless of internal forces between parts.
- Stability: An object is stable when the center of mass is above the base of support. Tipping happens when the center of mass moves outside the base. This is why wider stances are more stable.
- Rotation: Free objects rotate around their center of mass. A thrown football spins around the point where it would balance on your fingertip.
- Aerospace: Aircraft and rockets must have their center of mass within specific limits for stable flight. A plane with the center of mass too far back becomes uncontrollable.
- Astronomy: Binary star systems orbit around their common center of mass (called the barycenter). The Earth-Moon barycenter is about 4,600 km from Earth's center.
Continuous Bodies and Integration
For objects with continuously distributed mass (like a rod with varying density), the summation becomes an integral: x_cm = integral of x*dm / integral of dm. For a uniform body, the density cancels and the center of mass depends only on geometry. That is why the centroid of a uniform triangle is always at the intersection of the medians, regardless of the triangle's size or the material it is made from.
Composite Shape Method
Many real objects are combinations of simple shapes. A T-shaped beam, an L-bracket, or a plate with a hole can each be broken into rectangles, circles, and triangles. Find the centroid and area of each piece, then treat them as point masses. For a hole, use a negative mass (subtract the hole from the total). This composite method is the standard approach taught in engineering statics courses.
Common Mistakes to Avoid
- Confusing centroid (geometric center) with center of mass (mass-weighted center). They match only for uniform density.
- Forgetting that the formula gives each coordinate independently. Do not try to find the center of mass as a single distance.
- Using inconsistent units. All masses must be in the same unit and all positions must be in the same unit.
- Placing the origin inconsistently. The center of mass coordinates depend on where you put your coordinate system, though the physical location of the point does not change.