How to Calculate Center of Gravity

The center of gravity (CG) is the center of an object's weight distribution, where the force of gravity can be considered to act. It is the point in any object about which it is in perfect balance no matter how it is turned or rotated around that point. For a finite set of point masses, CG may be defined as the average of positions weighted by mass. That is, the (Sum of mass*position)/(Sum of mass).

Steps   [edit]

  1. A see-saw
    A see-saw
    Calculate the weight of the basic object. Let's use the example of two kids on a see-saw. The see-saw by itself weighs 30lbs. Since the see-saw is a symmetrical object, the CG of the empty see-saw will be exactly in the center of symmetry.
  2. Calculate the additional weights. In the example, assume there are two kids on the see-saw weighing 40lbs and 60lbs.
  3. Choose a starting point. This is called the datum. This point is arbitrarily placed at one end of the see-saw.
  4. Measure the distances from the datum to the center of each object. In the example, you must find the distances to the center of the see-saw and each of the two kids. The see-saw is 16ft long, so the center is 8ft from the datum. The kids are sitting exactly one foot from the end on either side, so their distances from the datum are 1ft and 15ft respectively.
  5. Multiply each distance by the respective weight. This gives you the moment for each object. First, the see-saw: 30lb * 8ft = 240ft*lb. The first kid: 40lb * 1ft = 40ft*lb. And the second kid: 60lb * 15ft = 900ft*lb. Add the moments to get 1180ft*lb for the total moment.
  6. Add the weights of all the objects. The sum of the weights are 30lbs + 40lbs + 60lbs = 130lbs.
  7. Divide the total moment by the total weight. 1180ft*lb / 130lb = 9.08ft. This is the distance from the datum to the center of gravity.


Tips   [edit]

  • To find the distance a person needs to move to balance the see-saw over the fulcrum, use the formula: (total weight) / (weight moved) = (distance CG moves) / (distance weight is moved). This formula can be rewritten to show that the distance the weight (person) needs to move equals the distance between the CG and the fulcrum times the weight of the person divided by the total weight. So the first kid needs to move -1.08ft * 40lb / 130lbs = -.33ft or -4in. (toward the edge of the see-saw).
  • To find the CG of a two dimensional object, use the formula Xcg = ∑xW/∑W to find the CG along the x-axis and Ycg = ∑yW/∑W to find the CG along the y-axis. The point at which they intersect is the center of gravity.
  • The definition for center of gravity of a general mass distribution is (∫ r dW/∫ dW) where dW is the differential of weight, r the position vector and the integrals are to be interpreted as Stieltjes integrals over the entire body. They can however be expressed as more conventional Riemann or Lebesgue volume integrals for distributions that admit a density function. Starting with this definition all properties of CG including the ones used in this article may be derived from properties of Stieltjes integrals.


Warnings   [edit]

  • Trying to blindly apply this mechanical technique without understanding the theory may result in errors.