# Centre of Mass: Rigid Bodies

### Definition of Centre of Mass

• The centre of mass of a rigid body is the point where the total mass of the body may be considered to be concentrated.

• It is a hypothetical or mathematical concept, as mass is typically spread throughout an object, not located at a single point.

• In a uniform gravitational field, the centre of mass is the same as the centre of gravity.

### Determining the Centre of Mass

• The coordinates of the centre of mass for a two-dimensional object are given by x̄ = (Σ(mx))/(Σm) and ȳ = (Σ(my))/(Σm), where m is the mass of each of the particles and (x,y) are their respective coordinates.

• For three dimensional bodies, the z-coordinate is often included, ẑ = (Σ(mz))/(Σm).

• If the body is symmetrical and made from a uniform material, the centre of mass is at the geometrical centre.

### Motion of a Rigid Body

• The motion of a rigid body can be seen as a combination of translation of the centre of mass and rotation about the centre of mass.

• Translational motion refers to the movement of the entire object in space without any rotation of the body itself.

• Rotational motion pertains to the movement of an object around a fixed axis. For a rigid body, this is about the centre of mass.

### Centre of Mass and Stability

• The position of the centre of mass relative to the points of support of an object is crucial for stability.

• If the line of action of the weight of a body passes through the base of support, the body is stable.

• An object with a lower centre of mass is generally more stable. For example, it’s harder to tip over a car with a low centre of gravity.

### Problem-solving Involving Centre of Mass

• When working on problems involving the centre of mass, always begin by identifying the system and locating the mass components.

• Problems may involve finding the centre of mass of composite bodies.

• It is important to understand how to apply the principles of the centre of mass to real-world situations, particularly in the fields of physics and engineering.

As a crucial element in Mechanics, understanding the centre of mass concept is instrumental in explaining how and why objects move the way they do. It enables us to predict the behaviour of physical systems and design stable and functional structures and vehicles.