What is the primary difference between the Centroid and the Centre of Gravity?

The Centroid is a purely geometric center based on the shape's area or volume, while the Centre of Gravity is the point where the weight acts. They only coincide if the object is made of a uniform, homogeneous material.

How does the calculation of CoG change when a part of the object is removed (e.g., a hole)?

The removed section is treated as a 'negative' area or weight. In the formula $\bar{x} = \frac{\sum A_i x_i}{\sum A_i}$, the area $A$ of the hole is subtracted in both the numerator and the denominator.

When does the Centre of Gravity NOT coincide with the Center of Mass?

They diverge when the object is in a non-uniform gravitational field, such as a structure so tall that gravity is significantly weaker at the top than at the bottom. For most engineering tasks on Earth, they are considered identical.

What is a common error when choosing a reference origin for CoG calculations?

Students often measure distances for different parts from different points. You must establish one fixed origin $(0,0)$ and measure all individual centroids $(x_i, y_i)$ relative to that same point.

What happens if you forget to subtract the 'negative area' from the denominator in a composite shape problem?

The total area (or weight) will be incorrectly high, leading to a CoG coordinate that is shifted too close to the origin or the hole. The denominator must represent the actual remaining material of the object.

Why is it incorrect to assume the CoG is always at the geometric center of an object?

This assumption only holds for homogeneous (uniform density) objects. If the object is composite (e.g., a hammer with a heavy head and light handle), the CoG will be biased toward the heavier end.

Define the Centre of Gravity in terms of moments.

It is the point about which the sum of the moments of the weights of all particles composing the body is zero for any orientation. Mathematically, it is the point where $W \bar{x} = \sum w_i x_i$.

What is the formula for the x-coordinate of the CoG of a composite area?

The formula is $\bar{x} = \frac{A_1 x_1 + A_2 x_2 + ... + A_n x_n}{A_1 + A_2 + ... + A_n}$, where $A_i$ is the area of each segment and $x_i$ is the x-coordinate of that segment's individual centroid.

State the principle of symmetry regarding the Centre of Gravity.

If a body possesses an axis of symmetry, its Centre of Gravity must lie on that axis. If it has multiple axes, the CoG is at their point of intersection.

Why does a low Centre of Gravity increase the stability of an object?

A lower CoG requires the object to be tilted at a much steeper angle before the vertical line of action of the weight falls outside the base of support. This creates a larger range of motion before toppling occurs.

Library Podcasts

Courses

Referral & Rewards

Revision Notes

AS-Level

Cambridge International Examinations

Physics

4. Forces, Density & Pressure

Centre of Gravity

Summary

The Centre of Gravity (CoG) is a fundamental concept in statics representing the unique point in a body where the entire weight is considered to act. It serves as the balance point of an object, where the sum of gravitational moments about any axis passing through it is zero, allowing complex distributed loads to be simplified into a single point force for equilibrium analysis.

1. Definition & Core Concepts

The Centre of Gravity (CoG) is defined as the point through which the resultant force of gravity acts for any orientation of the body. While gravity acts on every individual particle of an object, the CoG allows engineers to treat the entire body as a single point mass located at this specific coordinate.

In a uniform gravitational field, the Centre of Gravity coincides exactly with the Center of Mass, which is the point where the distribution of mass is balanced in all directions. This distinction is important in large-scale structures or varying gravitational fields, though they are treated as identical in most terrestrial engineering applications.

The concept of Weight ( $W$ ) is central here, where $W = mg$ . The CoG is the point where the total weight vector is applied, ensuring that the translational and rotational effects of gravity are correctly modeled.

Diagram showing a composite body consisting of a rectangle and a triangle, illustrating individual centers of gravity and the combined resultant center of gravity.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

It is vital to distinguish between the Centroid and the Centre of Gravity. The Centroid is a purely geometric property based on shape alone, whereas the CoG depends on the distribution of mass and the influence of gravity.

Feature	Centroid	Centre of Gravity
Basis	Geometry/Area/Volume	Weight/Mass Distribution
Dependency	Independent of material	Dependent on density ( $\rho$ )
Application	Pure mathematics/Fluid pressure	Statics/Stability/Dynamics

In most engineering problems involving uniform materials, the Centroid and CoG are at the same However, if a beam is made of two different materials (e.g., steel and wood), the CoG will shift toward the denser material, while the geometric Centroid remains unchanged.

5. Exam Strategy & Tips

6. Stability & Equilibrium

Centre of Gravity

Summary

1. Definition & Core Concepts

Diagram showing a composite body consisting of a rectangle and a triangle, illustrating individual centers of gravity and the combined resultant center of gravity.

2. Underlying Principles

The mathematical foundation of the Centre of Gravity is the Principle of Moments, which states that the moment of the resultant weight about any axis must equal the sum of the moments of the individual weights of the parts about that same axis. This is expressed as $W \bar{x} = \sum w_i x_i$ , where $W$ is total weight and $\bar{x}$ is the coordinate of the CoG.

For objects with continuous mass distribution, the summation is replaced by integration. The coordinates are found using $\bar{x} = \frac{\int x \, dW}{\int dW}$ , where $dW$ represents an infinitesimal element of weight located at position $x$ .

The Axis of Symmetry principle is a powerful logical shortcut; if a body has a line or plane of symmetry, the Centre of Gravity must lie on that line or plane. If a body has two axes of symmetry, the CoG is located at their intersection point.

3. Methods & Techniques

The Method of Composite Bodies is used for complex shapes that can be broken down into simpler geometric primitives like rectangles, triangles, and circles. The process involves calculating the area (or weight) and the individual CoG of each part, then applying the weighted average formula: $\bar{x} = \frac{A_1 x_1 + A_2 x_2 + ...}{A_1 + A_2 + ...}$ .

When dealing with Holes or Cutouts, the area of the removed section is treated as a 'negative area' in the composite formula. This subtracts the moment that would have been contributed by that material, effectively shifting the CoG away from the void.

For Non-Uniform Bodies, where density varies with position, the weight element $dW$ must be expressed in terms of density $\rho(x,y,z)$ and volume $dV$ . The formula becomes $\bar{x} = \frac{\int x \rho g \, dV}{\int \rho g \, dV}$ , requiring calculus to solve for the specific distribution.

4. Key Distinctions

Feature	Centroid	Centre of Gravity
Basis	Geometry/Area/Volume	Weight/Mass Distribution
Dependency	Independent of material	Dependent on density ( $\rho$ )
Application	Pure mathematics/Fluid pressure	Statics/Stability/Dynamics

5. Exam Strategy & Tips

Always Check Symmetry First: Before starting long calculations, look for axes of symmetry to immediately identify one or more coordinates of the CoG, which can save significant time and reduce calculation errors.
Establish a Consistent Origin: Choose a reference point (origin) that simplifies the math, such as the bottom-left corner or a line of symmetry. Ensure all distances ( $x_i, y_i$ ) are measured consistently from this single point.
Sanity Check the Result: The calculated CoG must always lie within the 'convex hull' of the object. If your result for a solid rectangle is outside the rectangle's boundaries, a sign error or division mistake has likely occurred.
Units and Signs: Ensure all areas or weights are in the same units. Remember that for 'negative areas' (holes), the area is subtracted in both the numerator (moments) and the denominator (total area).

6. Stability & Equilibrium

The location of the CoG relative to the Base of Support determines the stability of an object. An object is in stable equilibrium if a small displacement causes the CoG to rise, creating a restoring moment that returns the object to its original position.

An object will topple if its CoG moves outside the vertical projection of its base. This principle is why vehicles with a high CoG (like SUVs) are more prone to rolling over during sharp turns compared to low-profile sports cars.

Neutral Equilibrium occurs when the CoG remains at the same height regardless of displacement, such as a sphere rolling on a flat horizontal surface. In this state, the object stays in its new position without returning or toppling further.

Always Check Symmetry First: Before starting long calculations, look for axes of symmetry to immediately identify one or more coordinates of the CoG, which can save significant time and reduce calculation errors.
Establish a Consistent Origin: Choose a reference point (origin) that simplifies the math, such as the bottom-left corner or a line of symmetry. Ensure all distances ( $x_i, y_i$ ) are measured consistently from this single point.
Sanity Check the Result: The calculated CoG must always lie within the 'convex hull' of the object. If your result for a solid rectangle is outside the rectangle's boundaries, a sign error or division mistake has likely occurred.
Units and Signs: Ensure all areas or weights are in the same units. Remember that for 'negative areas' (holes), the area is subtracted in both the numerator (moments) and the denominator (total area).