Center of Gravity: Definition, Equation and Calculation

The center of gravity (not to be confused with center of mass) of a body is a point where the weight of the body acts and total gravitational torque on the body is zero.

To determine the center of gravity (CG) of an irregularly shaped body (say a cardboard), we take a narrow tipped object (say a sharp pencil). Now by trial and error, we can locate a point G on the cardboard, where it is balanced on the tip of the pencil. This point of balance is the center of gravity of the cardboard. The tip of the pencil at G provides the normal reaction R to the total weight mg of the cardboard. The cardboard is in translational equilibrium, as R = mg.

Fig. The resultant weight of the particles passes vertically through the CG of the body.

Also, the cardboard is in rotational equilibrium because if it were not so, the cardboard would tilt and fall due to an unbalanced torque. Force of gravity like m1g, m2g, m3g... etc. act on individual particles of the cardboard. They make up the torque on the cardboard. For its particle of mass mi, the force of gravity is mig. If ri is the position vector of this particle from CG of the cardboard, the torque about the CG is

CG of the cardboard is so located that the total torque on it to forces of gravity on all the particles is zero. Thus, total torque is:

As g is a non-zero value and same for all particles of the body, so the above equation can be written as

Equation satisfying center of gravity

This is the condition, when center of mass of the body lies at the origin. As position vectors are taken with respect to the CG, the center of gravity of the body coincides with the center of mass of the body.

However, if the body is so extended that g varies from part to part of the body, then the center of gravity will not coincide with center of mass of the body. For a body of small size, having uniform density throughout, the CG of the body coincides with the center of mass. In case of solid sphere, both CG and center of mass lie at center of the sphere. For a body of very large dimensions and having non-uniform density, the center of gravity does not coincide with the center of mass.

Calculation of Center of Gravity

Calculate the center of gravity of following diagram:

calculation of center of gravity of the system

Step 1: Draw Free Body Diagram of the System

Free body diagram of the system to calculate center of gravity

Step 2:  Find Weight Distance Moment with Reference to Datum

Datum is the arbitrary starting point on the end of the slab. Lets suppose, we choose point A as datum and find momentum with respect to that point. The total weight distance moment at point is given by:

Step 3: Calculation of Center of Gravity

To calculate the center of gravity, divide total weight distance moment by total mass of the system.

center of gravity is equal to weight distance moment divide by total mass

Thus, the center of gravity is 13 meter from left-hand side.

Shape of a Body and the Position of its Center of Gravity

The center of gravity of a body depends on the shape and size of the body. In a sphere, it is at the center. The CG of a body may not necessarily be within the body. For example, the CG of a ring does not lie in the material of the ring; it is at its geometric center. Similarly, in a hollow sphere, the CG is not within the material but lies at its geometric center.

The position of center of gravity changes with the change in shape of a body. An iron bar has CG at its middle point. But when the bar is bent into a circular ring, the CG is not within the material, it is at its geometric center. When any rigid body is supported at its CG, it is in equilibrium. The CG of different shape of bodies are given in the following table.

Center of gravity of different shape of body