What are the differences between the center of gravity and the centroid?

The centroid represents the geometric center of an object, whereas the center of gravity takes into account the distribution of weight throughout that object. In the case of uniform materials, the centroid and center of gravity will be the same point; however, for non-uniform materials, these two points may not align.

What factors influence the position of the center of gravity?

The position of the center of gravity (CG) is influenced by several factors, Mass Shape and Geometry Orientation

What is the method for calculating the center of gravity of a uniform object?

The center of gravity for simple shapes, such as rectangles and triangles, can be determined using their geometric properties. For instance, the CG of a rectangle is situated at its geometric center, whereas the CG of a triangle is found at the point where its medians intersect.

Is it possible for the center of gravity to lie outside the physical boundaries of an object?

Yes, the center of gravity (CG) can indeed be situated outside the physical boundaries of an object. This often happens with irregularly shaped objects or specific configurations. Examples: Gimbals and Gyroscopes, Tetrahedron

Understanding the Centre of Gravity: Key Concepts and Applications

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JEE Physics

Centre of Gravity

The centre of gravity (CG) is a key concept in physics and engineering, representing the point where an object's weight is evenly distributed in all directions. This point is essential for analyzing stability, balance, and motion across different systems. In uniform objects, the centre of gravity typically aligns with the geometric center. However, for irregular shapes, it must be determined through specific calculations. The CG significantly impacts how objects respond to various forces, including gravity, friction, and acceleration. A thorough understanding of the centre of gravity is vital for optimizing designs to enhance safety and efficiency.

1.0Centre of Mass

Centre of mass is a point where we can concentrate the whole mass of the body and it behaves in a similar manner as a point object behaves under the same circumstances.
The point in a system at which the whole mass of the system may be assumed to be concentrated for all translational effects of the system is called the Centre of Mass (CM).

2.0Properties of Centre of Mass

For symmetrical bodies having uniform distribution of mass, it coincides with the centre of symmetry or geometrical centre.

For a given shape it depends on the distribution of mass within the body and is closer to the massive part.

There may or may not be any mass present physically at the centre of mass and it may be within or outside the body.

Centre of Mass of Discrete Mass system

Assume that there are ndiscrete particles with position vector $r_{1}, r_{2}, \dots\dots r_{n}$ respectively.

$r_{1} = x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k}$

From COM definition of COM, Position Vector of centre of mass of all n particle

$r_{c m} = \frac{\sum m _{i} r _{i}}{\sum m _{i}} = \frac{m _{1} r _{1} + m _{2} r _{2} + m _{3} r _{3} + \dots\dots\dots m _{n} r _{n}}{m _{1} + m _{2} + m _{3} + \dots\dots\dots\dots\dots\dots\dots m _{n}}$

Centre of Mass of Two Particle System

Consider two particles of masses $m_{1}$ and $m_{2}$ with position vectors $r_{1}$ and $r_{2}$ respectively.

Centre of Mass of Continuous Mass Distribution

If a system has continuous mass distribution, treating the mass element d m at position $r$ as a point mass and replacing summation by integration:
$r_{c m} = \frac{\int r d m}{m}$ where $m = \int d m$
$x_{CM} = \frac{\int x d m}{\int d m} = \frac{\int x d m}{m}$
$y_{CM} = \frac{\int y d m}{\int d m} = \frac{\int y d m}{m}$
$z_{CM} = \frac{\int z d m}{\int d m} = \frac{\int z d m}{m}$

3.0Centre Of Gravity

It is the point in the body through which the resultant of all these parallel forces of attraction formed by weight of the body passes. It is usually denoted by C.G or Simply G.
Every body has one and one centre of gravity.

Body shows a total weight W, divided into three small parts of weight W1,W2,W3.

Each of the smaller components experiences a vertical pull due to gravitational forces. These pulls can be considered a parallel force system. This system will have a resultant force R, which is equal in magnitude to the sum of all the individual forces and acts downward along a different line of action.

4.0Determining Centre of Gravity

Balancing Method

Identify a Pivot Point: Find a point on the object where it can balance (e.g., edge of a table).
Balance the Object: Place the object on the pivot and adjust until it is stable.
Mark the Vertical Line: Use a plumb line or ruler to drop a vertical line from the pivot, indicating gravitational pull
Repeat: Move the object and find another pivot point; repeat the balancing process.
Locate the CG: The intersection of the vertical lines from both pivot points indicates the center of gravity.

5.0Centre of Gravity of Regular Objects

6.0Centre of Gravity of Solids Formula

The weight of a body acts as a force at its center of gravity, directed toward the center of the Earth. The position of the center of multiple bodies with weights $W_{1}, W_{2}, W_{3}$ etc. can be determined similarly to how the resultant of parallel forces is calculated.

$\overset{x}{ˉ} = \frac{Σ W x}{Σ W}$ ,

$\overset{y}{ˉ} = \frac{Σ W y}{Σ W}$ ,

$\overset{z}{ˉ} = \frac{\sum W z}{\sum W}$

If all the bodies are of the same material having density \rho,

Then $W_{1} = ρ V_{1}, W_{2} = ρ V_{2}, W_{3} = ρ V_{3}$

$\overset{x}{ˉ} = \frac{\sum ρ V x}{\sum ρ V} = \frac{\sum V x}{Σ V}$ ,

$\overset{y}{ˉ} = \frac{\sum ρ V y}{\sum ρ V} = \frac{\sum V y}{\sum V}$ ,

$\overset{z}{ˉ} = \frac{\sum ρ V z}{\sum ρ V} = \frac{\sum V z}{\sum V}$

If the bodies are made of the same material and have uniform density throughout, their center of gravity coincides with their center of volume. For bodies with the same cross-section but varying lengths, the center of gravity will still align with their center of volume.

$V_{1} = a l_{1}, V_{2} = a l_{2}, V_{3} = a l_{3}$ ……..etc

$\overset{x}{ˉ} = \frac{\sum a l x}{\sum a l} = \frac{\sum l x}{\sum l}$ , $\overset{y}{ˉ} = \frac{\sum a l y}{\sum a l} = \frac{\sum l y}{\sum l}$ , $\overset{z}{ˉ} = \frac{\sum a l z}{\sum a l} = \frac{\sum l z}{\sum l}$

7.0Sample Question on Centre of Gravity

Q-1. Classify between Centre of Mass and Centre of Gravity?

Solution:

Centre of Mass	Centre of Gravity
The centre of mass is the point in a body where its mass is evenly distributed, representing the average position of all its mass.	The centre of gravity is the point where an object's weight acts, resulting in zero net gravitational torque.
The centre of mass is independent of the gravitational field and is determined solely by the object's shape and mass distribution.	The centre of gravity relies on the gravitational field; in a uniform field, like near Earth’s surface, it aligns with the centre of mass.

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