Centre of Gravity

The centre of gravity (CG) of an object is the point at which the weight of the object acts and where the force of gravity can be considered to be concentrated. For a uniform gravitational field, the centre of gravity coincides with the centre of mass. However, in a non-uniform gravitational field, these two points can be different.

Understanding Centre of Gravity

The centre of gravity is an important concept in physics because it helps us predict the behavior of an object when subjected to gravitational forces. It is particularly useful in the fields of engineering and mechanics, where it is essential to know how objects will balance, remain stable, or move under the influence of gravity.

Properties of Centre of Gravity

• It is the point where the total weight of the body is thought to be concentrated.
• It is the point through which the resultant of gravitational forces on individual particles of the body passes.
• For symmetrical objects with uniform density, the centre of gravity is located at the geometric centre.
• The position of the centre of gravity can change if the shape or distribution of mass within the object changes.

Formulas Related to Centre of Gravity

The position of the centre of gravity can be calculated using the following formulas:

For discrete masses:

$$ x_{CG} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{CG} = \frac{\sum m_i y_i}{\sum m_i}, \quad z_{CG} = \frac{\sum m_i z_i}{\sum m_i} $$

For continuous bodies:

$$ x_{CG} = \frac{1}{M} \int x dm, \quad y_{CG} = \frac{1}{M} \int y dm, \quad z_{CG} = \frac{1}{M} \int z dm $$

Where:

• $x_{CG}, y_{CG}, z_{CG}$ are the coordinates of the centre of gravity.
• $m_i$ is the mass of the $i^{th}$ particle.
• $x_i, y_i, z_i$ are the coordinates of the $i^{th}$ particle.
• $M$ is the total mass of the body.
• $\int dm$ represents the integral of the mass distribution over the body.

Examples

Example 1: Simple Pendulum

Consider a simple pendulum consisting of a string and a bob. The centre of gravity of the bob is at its geometric centre. When the pendulum swings, it oscillates about the pivot point, but the motion can be analyzed by considering the forces acting on the centre of gravity of the bob.

Example 2: Balancing a Ruler

If you place a ruler on your finger, you can balance it by finding its centre of gravity. For a uniform ruler, the centre of gravity is at the midpoint. If you add a mass to one end of the ruler, the centre of gravity shifts towards that end, and you would need to adjust your finger position to balance it again.

Table: Differences Between Centre of Gravity and Centre of Mass

Centre of Gravity	Centre of Mass
Depends on the gravitational field.	Independent of the gravitational field.
May not coincide with the centre of mass in a non-uniform gravitational field.	Always represents the average position of mass distribution.
Relevant for analyzing stability and balance under gravity.	Relevant in all fields of mechanics, not just under gravity.
Can change with the orientation of the object in a gravitational field.	Remains constant for a given object unless its mass distribution changes.
For small objects on Earth's surface, it is practically the same as the centre of mass.	Used in the analysis of orbital mechanics and systems with no gravity.

Conclusion

The centre of gravity is a fundamental concept that helps us understand how objects will behave under the influence of gravity. It is crucial for designing stable structures, understanding the motion of bodies, and analyzing various physical systems. By knowing the centre of gravity, one can predict the stability and balance of objects and engineer solutions that are safe and reliable.