What is the primary difference between the Centre of Mass (CM) and the Centre of Gravity (CG)?

The CM is determined solely by the distribution of mass within an object, while the CG is the point where the weight acts and depends on the gravitational field. They only coincide perfectly in a uniform gravitational field.

How does the stability of a racing car compare to a standard SUV in terms of Centre of Mass?

A racing car is more stable because it is designed with a much lower centre of mass and a wider wheel track (base). This ensures that even during high-speed turns, the vertical line from the CM stays within the base, preventing toppling.

When comparing a uniform rod and a non-uniform rod of the same mass, where will their Centres of Mass lie?

The uniform rod's CM will be exactly at its geometric midpoint due to symmetry. The non-uniform rod's CM will be shifted toward the end with the higher linear mass density.

Is it possible for the centre of mass of a solid object to lie outside the object's physical boundaries?

Yes, the centre of mass is a mathematical point that depends on mass distribution and can exist in empty space. Common examples include a wedding ring, a hollow sphere, or a high jumper performing the Fosbury Flop.

What is a common mistake when identifying the CM of a composite object made of different materials?

A common error is assuming the CM is at the geometric center (centroid). If the materials have different densities, the CM will always be biased toward the denser material, regardless of the shape's symmetry.

Why is it incorrect to say an object topples because its 'Centre of Mass is too high'?

Height alone does not cause toppling; toppling occurs when the vertical line from the CM falls outside the base. A high CM is stable if the base is wide enough, and a low CM can topple if the base is narrow or the tilt is extreme.

Define the 'Centre of Mass' of a system.

The centre of mass is the unique point in a system where the sum of the weighted relative positions of the distributed mass is zero. It is the point that moves as if all the system's mass were concentrated there and all external forces were applied there.

What is a 'Plumb Line' and how is it used to find the CM?

A plumb line is a string with a weight attached that hangs perfectly vertical due to gravity. When an object is suspended, the CM always settles directly below the pivot point, so the plumb line marks a path that must contain the CM.

State the formula for the x-coordinate of the Centre of Mass for a discrete system of particles.

The formula is $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$, where $m_i$ is the mass of each particle and $x_i$ is its position relative to the origin. This represents the mass-weighted average position.

Why does an object suspended from a single point always come to rest with its CM directly below the pivot?

This occurs because the weight of the object acts through the CM. If the CM is not directly below the pivot, the weight creates a torque (moment) that rotates the object until the CM and pivot are vertically aligned, reaching equilibrium.

Centre of Mass | OCR AS-Level Physics

Centre of Mass

Summary

The centre of mass is a fundamental concept in mechanics representing the unique point where the entire mass of a system or object can be considered concentrated for the purpose of analyzing translational motion. It serves as the balance point of an object and is critical for determining stability, equilibrium, and the effects of external forces on complex bodies.

1. Definition & Core Concepts

The Centre of Mass (CM): This is the theoretical point at which the total mass of an object or system is thought to be concentrated. When a force is applied directly through this point, the object will undergo linear acceleration without any rotation.
Mass Distribution: The location of the CM depends entirely on how mass is distributed within the object. In a system of discrete particles, the CM is the weighted average of the positions of all individual masses.
Hypothetical Nature: It is important to recognize that the centre of mass is a mathematical construct and does not necessarily have to reside within the physical material of the object. For example, the centre of mass of a hollow ring or a horseshoe lies in the empty space at its geometric center.

Diagram showing stability: a stable block with its weight vector acting through the base, and a toppling block where the weight vector from the CM falls outside the base.

2. Underlying Principles

Symmetry Principle: For any object with a uniform density (homogeneous) and a clear geometric symmetry, the centre of mass must lie on the lines or planes of symmetry. For instance, the CM of a uniform sphere is at its geometric center, and the CM of a uniform rod is at its midpoint.
Gravitational Interaction: While the centre of mass is an intrinsic property of the object's mass distribution, the Centre of Gravity (CG) is the point where the weight of the object acts. In a uniform gravitational field, such as near the Earth's surface, the CM and CG are effectively at the same
Principle of Moments: The centre of mass is the point about which the sum of the moments of the individual mass elements is zero. This means that if an object is supported exactly at its CM, it will remain in perfect rotational equilibrium.

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

Verify the Base: When solving stability problems, always draw a vertical line straight down from the centre of mass. If this line passes through the area of the base in contact with the ground, the object is stable; if it falls outside, the object will topple.
Check for Uniformity: Exams often specify if an object is 'uniform.' If it is, you can immediately place the CM at the geometric center. If it is 'non-uniform,' you must use the principle of moments or given mass ratios to find the offset.
Sanity Check: Always evaluate if the calculated CM position makes physical sense. For example, in a system of two unequal masses, the CM must always be closer to the heavier mass than the lighter one.

6. Common Pitfalls & Misconceptions