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How to Calculate Center of Gravity

Accurate calculations of the center of gravity of a moving object are imperative to ensure its safety. There are several methods to find the center of gravity of an object, which must be calculated with the utmost accuracy.

The accuracy of center of gravity calculations depends on the object’s intended use. In the case of spacecraft, aircraft and other entities that rely on such calculations to perform, accurate testing equipment is imperative.

What Is Center of Gravity (CG)?

The center of gravity location is the average location of all the weight of an object. Archimedes of Syracuse discovered it over 2,000 years ago, but what is CG in measurement?  In simple terms, the center of gravity is the balance point of an object, also expressed as the point where all the mass appears to be located.

It’s a theoretical point at which the total weight of an object is thought to be concentrated. If, for example, the object in question is a perfect cube, the center of gravity would be in the exact center of that cube.

Gravity acts on all objects, pulling them down to the Earth’s center. The center of gravity is where an object would be in perfect balance. In the case of an airplane, the center of gravity would be where it would balance if it were possible to hang at that point.

What is the importance of determining the center of gravity location?

The center of gravity predicts the behavior of a moving body, like an aircraft when acted upon by gravity. This theoretical point simplifies complex calculations by treating the mass of an object as though it is concentrated in a single point.

Aside from moving objects, the center of gravity is also essential when designing static structures, such as bridges and buildings. Asymmetric objects made from different materials present more of a challenge, as the center of gravity is likely to be located some distance from its geometric center.

It’s critical to have accurate knowledge of an object’s center of gravity location to control the flight of that object. To ensure such objects remain balanced — and therefore safe — during motion, calculations of their center of gravity must be accurate.

The center of gravity location has several unique properties:

  • An object in space rotates around its center of gravity.
  • A force applied to the center of gravity causes pure translation.
  • The location of an object’s center of gravity is a critical parameter in determining its flight characteristics.

For example, for a straight flight, it’s essential to align a rocket motor’s center of thrust so that it aligns directly through the rocket’s center of gravity. The center of gravity affects the balance and stability of objects. For objects sitting on the ground, the lower the center of gravity, the more balanced and stable the object.

The center of gravity height is an essential parameter in the automotive industry. A vehicle’s center of gravity should be lower if stability is the focus. Vehicles with high CG, such as SUVs, are more likely to roll. Race cars have such low CG to the point that some organizations have rules around the allowed height of their center of gravity.


How do you find the Center of Gravity?

Where is the center of gravity located? It can be difficult to find the center of gravity of an object. Center of gravity is generally calculated first, using CAD models, for example. But even modern tools have their limitations.

Software can’t accurately determine the position of components such as cables. The way cables are routed inside the object can shift its center of gravity by a significant amount. Manufacturing tolerances also create uncertainties in the location of every payload component. All these small errors add up to a large center of gravity uncertainty.

This accumulation of errors explains why the center of gravity of an actual object must often be measured. A simple method of calculating the center of gravity of any object includes the following steps:

  1. Define a coordinate axis system for the object.
  2. Divide the object into smaller parts to determine the CG quickly. Consider more uniform shapes, such as triangles and rectangles, where possible.
  3. Using your predetermined axes, determine and express your CG in coordinates.
  4. Arrange all the numbers and coordinates to perform the calculation.

The formula to calculate the center of gravity is CoG = (ΣD* W) / ΣW. In words, this formula is — the location of the center of gravity can be found by summing (Σ) the multiplication of the distance by the weight and dividing it by the summation of all weights.

The location of the center of gravity is represented in coordinates:

  • The x-direction of the formula can be written as CoGx = (ΣDx* W) / ΣW.
  • The y-direction of the formula can be written as CoGy = (ΣDy* W) / ΣW.


Where Is the Center of Gravity Located in the Human Body?

Assuming the human body is completely still and standing, the center of gravity is above the pivot point in the hips. However, human beings don’t remain in a fixed position, so our center of gravity is fluid and changes as we move.


How to Calculate the Center of Gravity

There are different methods to calculate the center of gravity. Mathematically, center of gravity can be defined as the weighted average of all point masses in an object, which satisfies the following:

CG = sum(mi * ri)/sum(mi): Where ri is the position of each point mass and mi is the mass of each point mass

Rcg = the distance from the center of gravity

The first step in calculating the center of gravity of an object is to determine its weight. Regarding symmetrical objects, the center of gravity is always at the object’s center. The center of gravity is critical for objects in motion, as they must remain balanced.

For example, calculating the center of gravity of an aircraft may include the following steps:

  1. Determine the aircraft’s weight and all associated masses, such as wings, passengers and fuel.
  2. Choose an arbitrary starting point placed at one end of the aircraft, known as the datum.
  3. Measure the datum’s distance from the center of the main object and include any additional weights in your calculation.
  4. Multiply the object’s distance from the datum by its weight to give you the movement for each object.
  5. Add up the movements of all the relevant objects.
  6. Add the weights of all the objects.
  7. Divide the total movement by the total weight to give you the distance from the datum to the object’s center of gravity.


How Do You Measure the Center of Gravity?

Several concepts are used to measure the center of gravity of a real object, the simplest of which is mechanical. If we balance an object using string or rope, the point at which the object is balanced is the center of gravity. While this may be a simple method, it’s only sometimes practical or possible to suspend or balance objects for testing in this manner.

Another simple concept is to use a load cell system. This technique is used in the WCG Series and SE90168 Series of the center of gravity measurement instruments. The object is placed on a fixed platform connected to three load cells. The weighted average of each load cell reading gives the object’s center of gravity location.

A different concept is used where high accuracy is required. The object is placed on top of a table that pivots about a defined axis. The moment due to center of gravity offset of the object from the center axis is measured using a force transducer. The location of center of gravity is derived from the moment measurement by applying the following formula:

  • M = W x d: Where M is the moment applied, W is the weight of the object, d is the distance from the pivot point to the center of gravity of the object.

While creating your own center of gravity instrument is possible, calculations require more accuracy. For accurate calculations, a professional machine is more reliable. This moment measurement method is used in our KSR and SE8913 series of center of gravity measurement instruments.


Accuracy of Center of Gravity Measurement

Your requirement for accuracy will vary depending on what you’re trying to do. For example, if you want to stack books on a table, you only need to know the center of gravity within a few inches. But if you want your model airplane to fly straight, you need to know its center of gravity within a fraction of an inch.

In the case of aircraft and spacecraft, the accuracy requirements for center of gravity measurement leaves little room for error. The slightest mistake can significantly affect the aircraft’s ability to perform and may even be dangerous for pilots and other personnel. Below is a list of examples with typical accuracy requirements:

  • Human: 2 inches (5 cm)
  • Golf ball: 0.05 inch (1.25 mm)
  • Rocket: 0.01 inch (0.25 mm)
  • Spacecraft: 0.001 inch (0.025 mm)


Center of Gravity Testing With Raptor Scientific

Any high-performance vehicle that flies requires accurate knowledge of its mass properties. Raptor Scientific is a family of companies that manufactures state-of-the-art instruments to test aviation, space and military crafts, projectiles, and other objects that need to travel in a predictable pattern.

With unmatched accuracy and longevity, our mass properties testing apparatus and over 50 years of experience, Raptor Scientific can find the right solution to your unique challenge. Join the United States and international military and space agencies that leverage Raptor Scientific’s unmatched expertise and knowledge on mass properties measurement.

Contact us today to learn more, or shop our measurement tools here.