Centre of Mass and its Motion

A body’s or system of particles’ centre of mass is defined as a point where the body’s or system of particles’s masses appear to be concentrated. The centre of mass (also known as the point of equilibrium) is a location at the centre of the mass distribution in space where the weighted relative position of the distributed mass has zero-sum. In simple words, the centre of mass is the location of an object where we can assume the whole mass is centred. It is the average position of all parts of a system or mass distribution in space. 

A force is frequently applied now, resulting in linear acceleration without any rotational acceleration. When studying the dynamics of motion of a system as a whole, we ideally do not pay much attention to the dynamics of the individual particles of the system.

Determining the centre of mass:

Using gravity forces, we can empirically discover a body’s centre of mass if necessary. A body with asymmetrical mass and constant density will have its centre of mass at some distance from the geometric centre. In the same way, a spherically symmetric body with 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 centre for symmetric objects.

System of particles and the 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 gravity

Gravity is commonly thought to be a constant force operating on a body. The centre of gravity is the hypothesised point at which gravity acts on a body. As a result, the centre of gravity and mass are at the same spot only in a uniform gravitational field. The terms “centre of gravity” and “centre of mass” are used interchangeably in physics literature. Moreover, they refer to the same object.

The motion of the centre of mass

Consider the case of a multi-particle system. Every particle in that system moves at a different speed. What would be the best way to assign a velocity to the entire system? 

Consider the following system of particles: m1, m2, m3, and so on. These particles’ initial position vectors are r1, r2, r3,…rn. These particles have now begun to move in the direction of their position vectors. The objective is to determine the velocity and direction of the system’s centre of mass.

Whenever the external force on a system of particles is zero, Fext =0, and p is constant. According to the law of conservation of momentum, if the total external force acting on a system of particles is zero, the system’s linear momentum is constant.

Both translational and rotational motion can occur in a rigid body. It’s easier to work using a reference frame attached to the system’s centre of mass in these situations.

The centre of mass of a system’s velocity and acceleration is calculated in the same way as the centre of mass:

vCM= (m1v1 + m2v2 +……..+mnvn )/M

aCM= (m1a1 + m2a2 +……..+mnan)/ M

The advantage of utilising the centre of mass to analyse a system’s motion is that it behaves exactly like a single particle:

P = MvCM

F = MaCM

Conclusion

The centre of mass is a single point on a structure that describes an object’s motion if it reduces to a point mass. The significant property of the centre of mass is that it appears to carry the body’s entire mass. The terms centre of mass and centre of gravity are frequently used interchangeably only in uniform fields. The centre of mass is the average position of an object’s mass. There’s also the centre of gravity, where gravity appears to act.