Center of Gravity Calculator

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Center of gravity calculations is very important in most classical mechanics and engineering calculations. So, Mechanicalland provides you with a very easy-to-use center of gravity calculator that you can use in various fields.

And also, you just need to follow the instructions to calculate your center of gravity.

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What is the Center of Gravity Calculator?





In engineering, we can deal with both 2D and 3D calculations. Also, 2D calculations require only 2 coordinates to obtain the center of gravities of masses or mass systems. But 3D calculations require 3 coordinates X, Y, and Z. 

Calculation the center of gravity of mass systems is not a hard business with hand. But for a very high number of masses, it can be troublesome and time-consuming. So, you can use this center of gravity calculator to calculate the complex center of gravity problems. 

How to Use the Center of Gravity Calculator?

Firstly, you need to select 2D or 3D options. If you are dealing with 2D plane masses, you need to select 2D. And if you are dealing with 3D space masses, select 3D. 

  • If You Select 2D; Enter the number of masses that your system has. You can also divide a complex-shaped system into more basic multiple masses to calculate the center of mass of that system. You need to enter, how many masses you have in your system. 

Then you need to enter the mass, X, and Y coordinates of the center of gravities of these sub-masses respectively. If you click on the 'Calculate!' button, the center of gravity will be calculated. 

After that, the Center of gravity will be given in X and Y coordinates. 

  • If You Select 3D; Actually, the difference in the 3D center of gravity calculation is the additional Z coordinate. You need to make the same steps with 2D with an additional Z coordinate. 

So the Center of gravity will be given in X, Y, and Z coordinates. 

Center of Gravity Formula

The general form of the center of gravity formula is very simple. The general form of the center of gravity formula is like this; 

Center of gravity formula.

These center of gravity formulae are defining the calculation of the center of gravities of multiple objects in 3D space. Each center of gravity formula is for one coordinate. 

The numerator side of the center of gravity formula is the calculation of the effect of each mass on the system to the center of gravity on that coordinate. To find these effects, the mass of each object is multiplied by the coordinate. For example in the X coordinate, each mass is multiplied by its X coordinate value and all of them are summed. 

This summation is divided into the summation of the masses of the system. So with this calculation of the center of gravity formula, the total system's center of gravity in X coordinate is found. 

The calculation above is made in three coordinates and all three coordinates of the center of gravity of the system are calculated with the center of gravity formula. 

  • Center of Gravity of 2D Objects: For the 2D objects, the only difference is that there is no Z coordinate for the calculations. Only two centers of gravity formulas are used which are X and Y. 
  • Center of Gravity Calculator: According to the center of gravity formulas that we explained, we created a calculator which is called the center of gravity calculator. The use of the center of gravity calculator is very simple. 

Center of Gravity of the 3D Objects?

Instead of 2D objects, 3D objects are used in most of the machinery. 2D objects are generally used in the simplification of the calculations for which the 3rd dimensions of parts are the same.

Take a look at the very basic example below. Here, we have three objects named 'A', 'B', and 'C'. The information about the masses and positions in the coordinate systems of these objects is like this; 

Object A: (x, y, z) = (2, 5, 7); Mass 7kg. 

Object B: (x, y, z) = (-3, 5, -6); Mass 3kg. 

And Object C: (x, y, z) = (0, -8, 2); Mass 10kg. 

Think about the machinery that is constituted by these objects or parts. And we would like to calculate the center of gravity of this machinery. First of all, we need to calculate the center of gravities in each coordinate. To do it, we need to multiply the masses of each object with the X coordinate of each object. Then divide this calculation by the total mass of the objects to find the center of gravity in X coordinates.

Gx = (Ma*Xa + Mb*Xb + Mx*Xc)/(Ma+Mb+Mc); 

Gx = (7*2 + 3*(-3) + 10*0)/(7+3+10); 

And Gx = 0.25. 

The total center of gravity of the system is 0.25 in X coordinates. 

Let's do the same calculation for Y and Z coordinates. 

Gy = (Ma*Ya + Mb*Yb + Mx*Yc)/(Ma+Mb+Mc); 

Gy = (7*5 + 3*5 + 10*(-8))/(7+3+10); 

And Gy = -1,5. 

So the total center of gravity of the machinery in the Y coordinate is -1.5. 

For Z coordinates; 

Gz = (Ma*Za + Mb*Zb + Mx*Zc)/(Ma+Mb+Mc); 

Gz = (7*7 + 3*(-6) + 10*2)/(7+3+10); 

And Gz = 2.55. 

The total center of gravity of the machinery in the Z coordinate is 2.55. 

The total center of gravity of the three objects in 3D space is; 

G = (Gx, Gy, Gz) = (0.25, -1.5, 2.55). 

Gravity Center Calculation for 2D Objects

Gravity action.
Image Source: Engineersfield

2D objects are not real-life stuff. We use them in theoretical applications. We also use 2D objects as an approximation in calculations if the third dimension of the objects is the same. The Center of calculation of the 2D objects is simpler than the 3D objects, and because of this, we generally use it to make the calculations simple. 

Take a look at the simple example below about the center of gravity calculation for 2D objects. 

3 objects have names 'A', 'B', and 'C'. And because these objects are 2D, only X and Y coordinates are in the stage. 

Object A; (x, y) = (3, 5), Mass = 10kg. 

Object B; (x, y) = (-8, 4), Mass = 5kg.

And, object B; (x, y) = (2, 4), Mass = 3kg.

To find the center of gravity of these three objects combined, first, we need to find the center of gravity in X and Y coordinates separately. 

To do it, we need to multiply the X coordinates with the masses and sum up all the results. Then divide this result by the total mass. 

Gx = (Xa*Ma + Xb*Mb + Xc*Mc)/(Ma+Mb+Mc); 

= (3*10 + (-8)*10 + 2*3)/18; 

Gx = -2.44. 

Let's do the same calculation for the Y coordinates; 

Gy = (Ya*Ma + Yb*Mb + Yc*Mc)/(Ma+Mb+Mc); 

= (5*10 + 4*10 + 4*3)/18; 

Gy = 5.6. 

As a result, the total center of gravity of the 3 2D objects combined is; 

G = (Gx, Gy) = (-2.44, 5.6). 

The calculation for Two Objects

In the calculation of the center of gravity of two objects, first, we need to calculate the center of gravity of these objects separately. Think about a situation where we have two objects in space which has the coordinates and masses of;

A: (x, y, z) = (5, 10, 15), Mass = 10kg 

B: (x, y, z) = (-3, 3, 2), Mass = 20kg

To calculate the total center of gravity of these two masses, we need to find the total center of gravity of all the coordinates separately.

The gravity center of X coordinate; 

Gx = (Ma*Xa + Mb*Xb)/(Ma+Mb);

= 5*10 + (-3*20) = -10

Gx = -10/(10+20) = -0.33. 

In this calculation, first, we found the effects of all the masses on the place of the center of gravity on the X coordinate. We multiplied all the X coordinates with masses and summed up them. To find the exact place of the gravity center of the X coordinate, we divided this value by the total mass. And we can find Gx as -0.33.

Let's do the same thing for the Y coordinate.

Gy = (Ma*Ya + Mb*Yb)/(Ma+Mb);

= 10*10 + 20*3 = 160; 

Gy = 160/(10+20) = 5.33. 

We made the same calculation that we made for the X coordinates. We just changed the coordinates. And finally, let's calculate the gravity center in the Z coordinate.

Gz = (Ma*Za + Mb*Zb)/(Ma+Mb);

= 10*15 + 20*2 = 190; 

Gz = 190(10+20) = 6.33.

As a result, we can give the total center of gravity of these two objects in space as; 

Gtotal = (Gx, Gy, Gz) = (-0.33, 5.33, 6.33), 30kg. 

If there is a machinery that we constitute with these parts, the mass of this machinery will be 30 kg and its gravity center will be at -0.33, 5.33, 6.33 coordinates. 

Conclusion

Combined center of gravities of objects.
A very basic model of the center of gravity calculation method.

So, there is no limit to entering sub-mass numbers. Similarly, if you have a complex system to calculate the center of gravity, you can divide this system into more basic and geometrical masses to calculate the center of gravity of the whole system.

Above all, Mechanicalland does not accept any responsibility for calculations made by users in calculators. A good engineer must check calculations again and again.

Finally, you can find out many more calculators in Mechanicalland! Take a look at the other engineering calculators available in Mechanicalland!

Also, do not forget to leave your comments and questions below about the center of gravity calculator.

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Comments

2 responses to “Center of Gravity Calculator”

  1. Hiro Avatar
    Hiro

    Either I don’t understand how it works, or the CG calculator doesn’t work right.
    For 3 masses in 2D, in a system of reference, I have:
    Mass A: 2 kg, X=10m, Y=10m
    Mass B: 3kg, X=20m, Y=10m
    Mass C: 6kg, X=15m, Y=50m
    I put this in and I get the coordinates of the center of gravity:
    X=4.7222
    y=9.7222
    In the system of reference I used that doesn’t even fall inside the triangle formed by the three weights. It makes no sense.
    What’s the story?

    1. Uğur Uygun Avatar

      Hello Hiro,

      Thank you very much for warning us and providing detailed information.
      We found that there was an issue that we reviewed your alert for. The problem was that there was a problem with the inputs that we used through the browser, not the calculation. Thanks to your warning, we noticed this situation and fixed it.
      You can try the Center of Gravity Calculator operation again. Don’t forget to come back to us, smile 🙂

      As the Mechanicalland team, we thank you.

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