The center of gravity is the center of weight distribution of an object, the point where the force of gravity can be assumed to act. It is the point where the object is in perfect balance, no matter how it is turned or rotated around that point. If you want to know how to calculate the center of gravity of an object, then you need to find the weight of the object and all objects on it, locate the reference and insert the known quantities into the relative equation. If you want to know how to calculate the center of gravity, just follow these steps.
Steps
Part 1 of 4: Identify the Weight
Step 1. Calculate the weight of the object
When calculating the center of gravity, the first thing to do is find the weight of the object. Suppose we need to calculate the total weight of a 30 kg swing. Being a symmetrical object, its center of gravity will be exactly in its center if it is empty. But if the swing has people of different weights sitting on it, then the problem is a little more complicated.
Step 2. Calculate the additional weights
To find the center of gravity of the swing with two children on it, you will need to find their weight individually. The first child weighs 40 pounds (18 kg) and the second child weighs 60. We leave the Anglo-Saxon units of measure for convenience and to be able to follow the images.
Part 2 of 4: Determine the Center of Reference
Step 1. Choose the reference:
it is an arbitrary starting point placed on one end of the swing. You can place it on one end of the swing or the other. Let's assume the swing is 16 feet in length, which is about 5 meters. We put the center of reference on the left side of the swing, next to the first child.
Step 2. Measure the reference distance from the center of the main object, as well as from the two additional weights
Suppose the children are each seated 1 foot (30cm) away from each end of the swing. The center of the swing is the midpoint of the swing, at 8 feet, since 16 feet divided by 2 is 8. Here are the distances from the center of the main object and the two additional weights from the reference point:
- Center of the swing = 8 feet away from the reference point
- Child 1 = 1 foot from the reference point
- Child 2 = 15 feet from the reference point
Part 3 of 4: Calculate the Center of Gravity
Step 1. Multiply the distance of each object from the fulcrum by its weight to find its moment
This will allow you to get the moment for each individual item. Here's how to multiply the distance of each object from the reference point by its weight:
- The swing: 30 lb x 8 ft = 240 ft x lb
- Child 1 = 40 lb x 1 ft = 40 ft x lb
- Child 2 = 60 lb x 15 ft = 900 ft x lb
Step 2. Add the three moments
Just do the math: 240 ft x lb + 40 ft x lb + 900 ft x lb = 1180 ft x lb. The total moment is 1180 ft x lb.
Step 3. Add the weights of all objects
Find the sum of the weights of the swing, the first and the second child. To do this, you need to add up the weights: 30lb + 40lb + 60lb = 130lb.
Step 4. Divide the total moment by the total weight
This will give you the distance from the fulcrum to the center of gravity of the object. To do this, simply divide 1180 ft x lb by 130 lb.
- 1180 ft x lb ÷ 130 lb = 9.08 ft.
- The center of gravity is 9.08 feet (2.76 meters) from the fulcrum or 9.08 feet from the left side end of the swing, which is where the reference was placed.
Part 4 of 4: Verify the Result Obtained
Step 1. Find the center of gravity in the diagram
If the center of gravity you calculated is outside the object system, the result is wrong. You may have measured distances from multiple points. Try one more time with a new reference center.
- For example, in the case of the swing, the center of gravity must be anywhere on the swing, not to the right or left of the object. It doesn't necessarily have to be on a person directly.
- This is also true in two-dimensional problems. Draw a square large enough to include all objects related to the problem to be solved. The center of gravity must be within this square.
Step 2. Check the calculations if the result is too small
If you have chosen one end of the system as the center of reference, a small value puts the center of gravity right on one end. The calculation may be correct, but it often indicates an error. Did you multiply the weight and distance values together when you calculated the moment? That is the correct way of calculating the moment. If you added these values together, you will usually get a much smaller value.
Step 3. Solve if you have more than one center of gravity
Each system has only a single center of gravity. If you find more than one, you may have skipped the step where you add all the moments. The center of gravity is the ratio of total moment to total weight. You don't need to divide each moment by your weight, since that calculation just tells you the location of each object.
Step 4. Check the calculation if the obtained reference center differs by an integer
The result of our example is 9.08 ft. Suppose your test results in a value such as 1.08 ft, 7.08 ft, or another number with the same decimal (.08). This probably happened because we chose the left end of the swing as the center of reference, while you chose the right end or some other point at a full distance from our center of reference. Your calculation is in fact correct regardless of which center of reference you choose. You simply have to remember that the center of reference is always at x = 0. Here is an example:
- In the way we solved the center of reference is on the left end of the swing. Our calculation returned 9.08 ft, so our center is 9.08 ft from the center of reference on the left end.
- If you choose a new center of reference 1 ft from the left end, the value for the center of mass will be 8.08 ft. The center of mass is 8.08 ft from the new center of reference, which is 1 ft from the left end. The center of mass is 08.08 + 1 = 9.08 ft from the left end, the same result we calculated earlier.
- Note: When measuring a distance, remember that the distances to the left of the center of reference are negative, while those to the right are positive.
Step 5. Make sure your measurements are straight
Suppose we have another example with "more children on the swing", but one of the children is much taller than the other, or maybe one of them is hanging from the swing instead of sitting on it. Ignore the difference and take all measurements along the swing, in a straight line. Measuring distances on slanted lines will lead to close but slightly offset results.
As for problems with the swing, what you care about is where the center of gravity is along the right or left side of the object. Later, you may learn more advanced methods of calculating the center of gravity in two dimensions
Advice
- To find the two-dimensional center of gravity of the object, use the formula Xbar = ∑xW / ∑W to find the center of gravity along the x axis and Ycg = ∑yW / ∑W to find the center of gravity along the y axis. The point where they intersect is the center of gravity of the system, where gravity can be thought to act.
- The definition of the center of gravity of a total mass distribution is (∫ r dW / ∫ dW) where dW is the weight differential, r is the position vector and the integrals are to be interpreted as Stieltjes integral along the whole body. However they can be expressed as more conventional Riemann or Lebesgue volume integrals for distributions admitting a density function. Starting from this definition, all the properties of the centroid, including those used in this article, can be derived from the properties of the Stieltjes integrals.
- To find the distance at which a person must position themselves to balance the swing over the fulcrum, use the formula: (Child 1 weight) / (Child 2 distance from the fulcrum) = (Child 2 weight) / (Child 1 distance from the fulcrum).
