# Center of Mass

## Center of Mass

Centre of Mass

Defining Centre of Mass

• The centre of mass of a system is the point at which all mass could be considered to be concentrated for the purpose of calculating the system’s gravitational behaviour.
• It is the average position of all the mass that makes up a system and is typically located within the object.

Calculating Centre of Mass

• The centre of mass can be calculated by dividing the sum of moments (product of mass and distance from a chosen point) by the total mass of the system.
• For two-point masses, the centre of mass lies closer to the more massive point.

Motion of Centre of Mass

• The centre of mass of a system of particles moves as though all of the external forces were applied to it and all of the external torques were about it.
• The velocity of the centre of mass is the mass-weighted average of the velocities of the individual particles.
• The acceleration of the centre of mass is the result of net external forces only. It is independent of internal forces.

Properties of Centre of Mass

• The centre of mass can be a point in space not occupied by any particles of the system, like in a circular hoop or a hollow sphere.
• In the absence of external forces, the centre of mass moves along a straight line.
• In a collision or explosion, the centre of mass follows the same trajectory before, during and after the event.
• The centre of mass can be used to predict the rotation of a body about a fixed axis.

Influence of Gravity

• The gravitational force on a body can be considered to act at the body’s centre of mass.
• For bodies near the Earth’s surface, the weight w of the body acts vertically downward through the centre of mass, creating an effective gravity force.

Real-World Applications

• Our centre of mass is positioned such that we can stand and move without toppling over.
• In sports like gymnastics and diving, athletes adjust the position of their limbs to manipulate their centre of mass and perform complex movements.
• Vehicles’ design aims to keep the centre of mass low to prevent toppling over during manoeuvres.