Introduction to centre of mass

The centre of mass is the unique spot where the entire mass of an object or system of particles is concentrated. It is the point where force applied results in linear acceleration without an angular acceleration. The centre of mass of objects varies. For a single rigid body, it is fixed in relation to the body. In a body with uniform density, it will be located in the centroid. In hollow or open-shaped objects, the centre of mass may be located outside the body. A force applied to or against an object’s mass cannot rotate it. The centre of mass equation applied to the object’s left or right side will cause it to rotate. The gravitational pull is attributed to an object’s centre of mass. Due to its low centre of gravity, a mass rises when flipped. As the centre of mass is always below the point of hanging, we can hang a flat item to find its centre of mass.

Centre of mass formula

Even though the centre of mass and centre of gravity often coincide, they are different. When the entire system is subjected to uniform gravitational fields, both the centre of gravity and centre mass will be the same. The shape of the object affects the centre of mass formula too.

Centre of mass formula for pointer objects:

zcom =i=1kmizii=1kmi

mi is mass of ith object

zi is distance from the z-axis of ith object

Determining the centre of mass

Using gravity forces, we can empirically discover a body’s centre of mass if necessary. In part, this is possible because the centre of mass and the centre of gravity in the parallel gravity field at the Earth’s surface are both at the exact location. A body with asymmetrical mass and constant density will have its centre of mass on this axis. In the same way, a spherically symmetric body that has a constant density will have its centre of mass at the centre of the sphere’s axis. We can assume that the centre of mass will always be in the same place for any given symmetry.

System of particles and centre of mass

If a rigid body is viewed as a particle, we’ve just dealt with translational motion. However, when a stiff object rotates, the movement of its constituent particles is not uniform. However, as a system of particles, we must approach it as a firmly connected group of particles following the centre of mass formula.

Internal forces may be at play when it comes to particles or bodies that aren’t firmly attached. Particle systems are capable of complicated motion, but one location, known as a mass centre, is responsible for all translational motion in the system.

Centre of mass of a two-particle system

Let’s assume a system of two particles with masses m₁ and m₂. Both are present on points A and B, respectively. Assume r₁ and r₂ be the position vectors with respect to the origin O.

Thus, the centre of mass C of that particular system with the position vector r_cm is as follows:

r_cm = (m₁ r₁+m₂ r₂)/(m₁+m₂)

Example of the centre of mass

Let’s have a look at an example to illustrate the centre of mass:

Assume you have a one-yard rod of negligible weight. It has one ball at both ends. One of the balls weighs 6 pounds, whereas the other weighs 2 pounds. These balls are placed at (0, 0) and (3 ft, 0) respectively.  Now, where on the road will it be possible to balance the system?

As we have weight instead of masses, we know that the force of gravity will be the same for both objects here. Therefore, the centre of mass and the centre of gravity are the same in this question. This concept is, however, not applicable to some systems such as satellites or planetary bodies with eccentric orbits.

Using equation of centre of mass, we get:

x_cm = (6 lb)(0 ft)+(2 lb)(3 ft)/(6 lb+2 lb)

= (0 + 6)ft lb / 8 lb

=0.75 ft

Centre of gravity

The centre of gravity is the point at which the gravitational pull operates on an object or system. It’s the point at which the resulting torque from gravity forces vanishes into thin air. The centre of gravity and centre of mass is always the same in a uniform gravitational field. When referring to the same point or location, the words “centre of gravity” and “centre of mass” are commonly used interchangeably.

Conclusion

An object’s centre of mass is a spot where the entire mass of the object or system of particles can be seen to be concentrated. The centre of mass and centre of gravity are different terms. However, they interconnect. The centre of gravity is the point at which the gravitational pull operates on an object or system. The centre of gravity and centre of mass is always the same in a uniform gravitational field. The formula or equation of calculating varies based on a pointer or extended object. Using gravity forces, we can empirically discover a body’s centre of mass if necessary. The formula of the centre of mass for the two-particle system is

r_cm = (m₁ r₁+m₂ r₂)/(m₁+m₂)

Where m₁ and m₂ are masses of two objects and r₁ and r₂ are their position respectively.