Center of Mass Calculator

Created by Hanna Pamuła, PhD
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 20, 2022

With this center of mass calculator, you can quickly find out the center of mass of up to 10 discrete masses. What is more, the tool allows you to choose between 1-, 2- or 3-dimensional systems. Impressive, isn't it? Below you'll also find the center of mass definition, as well as information on how to find the center of mass of a triangle.

Apart from the physics lesson, the center of the mass equation may be useful in real-life applications:

  • if you're wondering how to find the center of mass when swinging with your child on the seesaw, you can make quick calculations with our tool
  • for the DIY enthusiast, the center of mass equation is super useful for making e.g., a crib or ceiling mobile
  • for those interested in cars, we also have the unique car mass center calculator!

Center of mass definition

The center of mass is the point at which the whole mass of the body or system can be considered to be concentrated. In other words, if you apply a force to that point, it will cause only linear acceleration and no angular acceleration. Don't forget that if the object has uniform density, then the center of mass is located at the centroid of the object. This fact can make your calculations way easier!

🙋 Hey, if you need a quick refresher on the notions mentioned above, try our acceleration calculator and angular acceleration calculator!

In one plane, you can imagine the center of mass as a lever or a seesaw, with a mass on each end:

Center of mass illustration in 1D. 2 masses shown.

If you have two objects with the same mass, the center of mass will be exactly half of the distance between them. But what happens if you have a more massive object at one end? Where's the pivot point? Intuitively, we know that the center of mass will be closer to a heavier object. Where exactly it lays, you can find out using the center of mass equation.

How to find the center of mass? Center of mass equation

In this center of mass calculator, the center of mass equation we've used is:

Center of mass = (m1r1 + m2r2 + ... + mNrN) / (m1 + m2 + ... + mN)

where

N - number of masses m

ri - distance from the reference point

The center of mass equation can be also shortened to:

Center of mass = 1/M * Σ miri

where M is the sum of the masses of all of the particles: M = m1 + m2 + ... + mN

That's the general formula, but you can also extend that center of the mass equation to 1,2 or 3 dimensions in the Cartesian coordinate system. For example, if you have N particles (up to 10 in our calculator) in 3D, you can calculate the center of mass using:

xcom = 1/M * Σ mixi

ycom = 1/M * Σ miyi

zcom = 1/M * Σ mizi

where i = 1,2, ..., N.

The formulas above work perfectly for discrete masses. However, if you have a continuous object with variable density, the computation of its integrals is needed.

Center of mass of a triangle

Center of a triangle illustration. Point where you can balance the triangle.

If you're searching for the center of mass of a triangle, we're pretty sure that you want to find the centroid of a triangle - the centroid of an object with uniform density. All you need to do is to calculate the arithmetic mean of the vertex coordinates. Given A = (x1,y1), B = (x2,y2), C = (x3,y3), the center of mass equation can be written as:

G = [ (x1+x2+x3)/3 , (y1+y2+y3)/3 ]

To calculate the center of mass of a triangle with this tool, enter the coordinates of the points of the triangle and make sure that the masses are all the same (the specific mass value doesn't matter). Also, you can simply use our centroid calculator.

Coordinate system. Equivalence of center of the mass of a triangle and equal massses in triangle's three vertices

While that is the most often common case, we cannot exclude the other cases: what happens if you have three masses forming the vertices of a triangle, but they differ in mass?

Let's assume that the mass in point B is 3 times bigger than the mass in points A and C:

Differences in the center of mass: equal massses in triangle's three vertices vs one of the masses is 3 times bigger than the others. Center of the mass moved towards heaviest mass.

As you can see, the center of mass moved towards the heavier point.

Center of mass vs. center of gravity

Center of mass and the center of gravity are the same when the gravitational field is considered to be uniform. So, if the gravity gradient effects are negligible (which is most cases), you can use both terms interchangeably. However, if you're looking at some large interstellar object, e.g., satellites and planets, a slight variation in the gravitational field may lead to torque, which would mean you need to differentiate between those two terms.

Hanna Pamuła, PhD
Dimensions
1D
Center of mass illustration in 1D. 2 masses shown
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