Understanding the Center of Mass and Center of Gravity

What is the Center of Mass?

The center of mass might sound complex, but it’s essentially the average position of all the parts of a system, taking their masses into account. Imagine it as the balancing point where the mass is equally distributed in all directions. For rigid bodies like a disc, the center of mass is fixed at a specific location, like the center of the disc.

Finding the Center of Mass

Determining the center of mass isn’t as daunting as it sounds. Simple experiments, like the table edge method or the plumb line method, can help identify this point. The table edge method works well for small objects with flat sides, while the plumb line method is useful for freely suspended objects.

Mathematical Calculation

Mathematically, you can find the center of mass by adding up the mass of each part of an object, multiplied by its position, and then dividing by the total mass of the object.

Center of Mass Formula
Center of Mass=∑(mass×position)∑massCenter of Mass=∑mass∑(mass×position)​

Benefits of Center of Mass

Knowing the center of mass is practical. It helps determine the maximum angle at which an object can tilt before toppling over. Additionally, it simplifies mechanics problems involving irregularly shaped objects by treating them as if all their mass is concentrated at the center of mass, acting like a point mass.

Center of Gravity

The center of gravity is where gravity’s force acts on an object. In many cases, especially in mechanics problems, the terms center of gravity and center of mass are used interchangeably because they often coincide. Yet, they aren’t precisely the same.

Locating the Center of Gravity

Finding the center of gravity involves summing the multiplication of the distance by the weight (area) and dividing it by the summation of all the weights (areas).

Center of Gravity Formula
Center of Gravity=∑(distance×weight)∑weightCenter of Gravity=∑weight∑(distance×weight)​

Differences between Center of Mass and Center of Gravity

Engineers, for instance, consider the center of gravity when designing vehicles. They aim to position it as low as possible to enhance handling. The image below illustrates how the center of mass and center of gravity might not always align.

Examples

Understanding through examples can make things clearer:

• The center of mass of a two-particle system lies on the line connecting the particles, closer to the more massive object.
• A solid triangle’s center of mass is at the centroid, the average of the three vertices.
• For a rectangle, the center of mass is at the intersection of the two diagonals.
• In a spherically symmetric body, like Earth, the center of mass is at the center, despite varying density.