Centre of Mass

Mass refers to the total quantity of matter in a physical body. It is distributed throughout a body and not located at a single point. However, to bring simplicity to physics calculations, it is assumed that all mass is concentrated at a single point within the body. If you can pinpoint the correct position within the body, the equations of motion give the same results as when you would calculate in a situation where the mass is spread out. The only difference is that centre of mass calculations are less complicated. This particular position is called the centre of mass. 

What is the centre of mass?

The centre of mass is an imaginary point where the entire mass of that object is said to be aggregated. It is a unique point at the centre of a distribution of mass in space. The property of this unique point is that the weighted position vectors relative to this point sum to zero. Another way to define the centre of mass is the mean location of a distribution of mass in space.

As we know, a rigid body consists of many particles. Each such particle has some mass. When put together, the mass of each such particle adds up to the total mass of that body. Now, this system of tiny masses can be considered as a system of parallel forces within the body. The centre of all these parallel forces is the centre of mass of that body.

How to Locate the Centre of Mass?

It is easy to locate the centre of mass in rigid objects of a simple shape. For them, the centre of mass is at the centroid. The centroid of any shape is a point where the shape’s cutout can balance perfectly on a pinpoint. 

• So, for a square, the centre of mass is located at the middle point where the two diagonals intersect. 
• Whereas for a ring, the centre of mass is located at the radial centre, where there is no material. 

Calculating the centre of mass for a system of particles:

To calculate the centre of mass for a system of particles,

Imagine a system of particles Pi, where i = 1, 2, 3, …, n

Each particle has a mass value of mi, where i = 1, 2, 3, …, n

And these particles are located in space with coordinates ri, where i = 1, 2, 3, …, n

The coordinates of the centre of mass are denoted by R, whereas the total mass of all particles is denoted by M.

In this situation, the coordinates of the centre of mass  satisfy the following situation, as we learned from the definition:

i=1nmi (ri – R) = 0

∴ R = 1M i=1n miri, 

where M = i=1nmi is the total mass of all the particles.

Centre of mass examples:

For practical purposes, the centre of mass and the centre of gravity can be treated as the same because every object on the Earth is naturally under the influence of gravity. Some centre of mass examples in everyday life are as follows:

• When loading an aeroplane, the centre of mass is always verified from tables or calculated afresh. If the centre of mass is not within limits, the authority of pitch control is likely to be lost. As a turn of events, the aeroplane will stall out and not leave the ground.
• One of the most common centre of mass examples is how it helps balance a boat in the water. It is used to define a point of lateral stability. Upon crossing this point, the boat will roll over into the water.
• Whenever you throw an object, the centre of mass is involved in the act. We usually find a balance point to throw it in the direction and motion that we have thought of. If the centre of mass would not come into play, the object could go in any random direction.

Conclusion:

The centre of mass is an important entity that helps calculate various entities for a body. It also finds use in our everyday lives through the centre of mass examples shown above. As you prepare for the topic through these centre of mass JEE notes, make sure to also go through the questions given below for a better understanding of the topic.