Centre Of Mass Formula, Overview, Principle, Equation

The centre of mass of a body or system of particles is defined as a place at which the entire mass of the body or all the masses of a particle system appear to be concentrated.

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The concept of centre of mass (COM) is crucial in physics and engineering. It is a place within or outside an object or system where the entire mass of the object can be concentrated, and the object behaves as if all its mass were concentrated at that point. For a single isolated object, the centre of mass is just the average position of all its constituent particles weighted by their respective masses.

What is Centre of Mass?

The centre of mass of a body or system of particles is defined as a place at which the entire mass of the body or all the masses of a particle system appear to be concentrated. In physics, the centre of mass is defined as a point in space at the centre of the distribution of mass (also known as the balance point), where the weighted relative position of the dispersed mass has a sum of zero. In layman’s terms, the centre of mass is a point relative to an item. It is the mean location of a distribution or the average position of all the pieces of the system.

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Determining of Centre of Mass

We can use gravity forces on the body to experimentally identify a body’s centre of mass. This is possible because the centre of mass and the centre of gravity in the parallel gravity field at the earth’s surface are the same. Furthermore, the centre of mass of a body with an axis of symmetry and constant density will be on this axis. Similarly, the centre of mass of a circular cylinder with constant density will be on the axis of the cylinder. If we consider a spherically symmetric body with constant density, its COM is located at the middle of the sphere. In principle, the centre of mass of anybody with any symmetry will be a fixed point of that symmetry.

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Centre of Mass Formula

The formula to calculate the centre of mass (COM) of a system of ‘n’ particles with masses β€˜π‘š1β€˜ π‘š2‘, …, ‘π‘šπ‘›Β 

‘, and coordinates (π‘₯1,𝑦1,𝑧1), (π‘₯2, 𝑦2, 𝑧2), …, (π‘₯𝑛 , 𝑦𝑛 , 𝑧𝑛 ), respectively, is as follows:

COMπ‘₯=π‘š1π‘₯1+π‘š2π‘₯2+…….+π‘šπ‘›π‘₯𝑛/π‘š1+π‘š2+…..β€ˆβ€ˆβ€ˆβ€ˆβ€ˆ+π‘šπ‘›

COM𝑦= π‘š1𝑦1+π‘š2𝑦2+…….+π‘šπ‘›π‘¦π‘›/π‘š1+π‘š2+…..β€ˆβ€ˆβ€ˆβ€ˆβ€ˆ+π‘šπ‘›

COM𝑧= π‘š1𝑧1+π‘š2𝑧2+…….+π‘šπ‘›π‘§π‘›/π‘š1+π‘š2+…..β€ˆβ€ˆβ€ˆβ€ˆβ€ˆ+π‘šπ‘›

Where:

COMπ‘₯, COM𝑦, COM𝑧

Β :The π‘₯Β , 𝑦 , and 𝑧 

Β coordinates of the centre of mass, respectively.

π‘š1,β€ˆπ‘š2,β€¦β€¦β€¦π‘šπ‘›

: The masses of the individual particles.

π‘₯1,β€ˆπ‘¦1,β€ˆπ‘§1,β€ˆπ‘₯2,β€ˆπ‘¦2,β€ˆπ‘§2,……π‘₯𝑛,β€ˆπ‘¦π‘›,β€ˆπ‘§π‘›β€ˆ

Β The π‘₯,β€ˆπ‘¦,Β  and 𝑧 

Β coordinates of each individual particle, respectively.

The centre of mass formula represents the weighted average of all particle positions, with the mass of each particle determining its contribution to the total position of the centre of mass. This formula helps us to locate the location in space where we can suppose the total mass of the system is concentrated. The concept of centre of mass simplifies complex system analysis and is essential in understanding the motion and behaviour of objects and systems in numerous domains of physics and engineering.

We can expand the formula to a particle system. The equation can be applied to each axis separately.

7separately

If we have point objects, we can utilise the calculation. If we need to identify the centre of mass of an extended item, such as a rod, we must employ a different approach. We must consider a differential mass and its position before integrating it throughout the entire length.

6entire length

Assume we have a rod, as indicated in the illustration, and we need to determine its centre of mass.

Assume that the entire mass of the rod is M and that the density is uniform. We also suppose that the width of the rod is insignificant, implying that the centre of mass is on the x-axis. We consider a tiny dx away from the origin. Therefore,

π‘‘π‘š=𝑀/𝑙 𝑑π‘₯

Using the equation to get the centre of mass,

5equation

When we integrate it from 0 to l, we obtain

π‘‹π‘π‘œπ‘š=𝑙/2

We may find the centre of mass for any geometrical shape using the method described above. You can experiment with a semi-circular ring or a triangle. As a result, if a force is applied to that stretched object, it can be considered to act through the centre of mass and so converted to a point mass.

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Centre of Mass of a Body Having Continuous Mass Distribution

If the given item is not discrete and their distances are not particular, the centre of mass can be calculated by considering an infinitesimal element of mass (dm) at x, y, and z distances from the origin of the chosen coordinate system.

π‘‹π‘π‘œπ‘š= ∫ π‘₯π‘‘π‘š/π‘‘π‘š

π‘Œπ‘π‘œπ‘š= ∫ π‘¦π‘‘π‘š/π‘‘π‘š

π‘π‘π‘œπ‘š= ∫ π‘§π‘‘π‘š/π‘‘π‘š

Centre Of Mass Formula FAQs

What is the centre of mass, and why is it important?

The centre of mass is a point within or outside an object or system where the entire mass of the object can be concentrated. It simplifies the analysis of complex systems and allows us to describe the motion and behaviour of objects as if all their mass were concentrated at that point. It is crucial in understanding equilibrium, collisions, and overall motion of objects and systems.

What is the significance of the centre of mass in the study of collisions?

In collisions between objects, the centre of mass remains at a constant velocity if no external forces act during the collision. This property is known as the conservation of momentum. The centre of mass simplifies the analysis of collision problems and helps determine the behaviour of the entire system.

Is the centre of mass always located within the object?

No, the centre of mass can be located both within and outside an object or system. It depends on the distribution of mass within the object. For irregularly shaped objects, the centre of mass might be outside the physical boundaries of the object.

How is the centre of mass related to stability?

In stable equilibrium, the centre of mass of an object or system is at its lowest possible position. If the centre of mass is raised above the base of support, the object becomes unstable and tends to topple over.

How is the centre of mass used in real-world applications?

The concept of the centre of mass is applied in various fields, such as physics, engineering, robotics, and astronomy. It helps in designing stable structures, analysing the motion of celestial bodies, predicting the behaviour of moving objects, and optimizing the design of vehicles and machines.

Can the centre of mass change over time?

Yes, the centre of mass can change if the distribution of mass within an object change or if external forces act on the object. However, in many scenarios, the centre of mass remains relatively constant, simplifying the analysis of motion and behaviour.

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