# Centre of Mass

## What is the Centre of Mass?

• The centre of mass of a system is its balance point when under the influence of gravity.
• Mathematically, it is defined as the vector sum of the products of the mass and position vector of each constituent part, divided by the total mass.
• When calculating, ensure to remember that the position vector is taken from a specific point, referred to as the origin.
• The origin can be anywhere in the space, but it is often convenient to take it as a specific point in the given problem for ease of calculations.

## Centre of Mass for Different Shapes

• For symmetrical shapes, the centre of mass is at the geometric centre.
• For irregular shapes, the centre of mass will not always be at the geometric centre. This is typically calculated by breaking the shape into smaller shapes for which the centre of mass is known.
• The centre of mass of a uniform rod lies at the midpoint.
• The centre of mass of a uniform circular disc or ring lies at the centre.
• For a triangle, it is located at the point where all three medians intersect, also known as the centroid.
• For a rectangle or square, the centre of mass lies at the intersection of the diagonals.

## Using Calculus to Find the Centre of Mass

• Calculus is often required to solve for the centre of mass when dealing with complicated or continuous distributions of mass.
• The integration method is used to calculate the centre of mass for complex shapes or for objects with a variable density.
• Remember that when using integration, you often need to evaluate the integral over the entire shape or distribution of mass.

## Implications and Applications

• Understanding the concept of the centre of mass is important for analysing the motion of objects, as it provides the point about which the object will rotate.
• It has wide application in the field of physics including, but not limited to, mechanics, orbital motion, and can be extended to the motion of large scale objects such as planets and galaxies.
• Knowledge of the centre of mass is a fundamental aspect of the design and analysis of structures and mechanical systems in engineering fields as well.