Centre of Mass Motion

Introduction

A rigid body system’s centre of mass is an important feature. Usually, more than one particle is present in these systems. It’s crucial to look at these systems as a whole. These bodies must be treated as a single point mass to perform mechanical calculations. A position like this is referred to as the centre of mass. Mechanical systems frequently move in a transient or rotatory fashion. In this situation, the centre of Mass moves, gaining velocity and acceleration. Let’s dive and learn in detail about the centre of mass motion.

Centre of Mass

A body’s or system of particles’ centre of mass is defined as a point where the body’s or system of particles’ 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 a zero-sum. In simple words, the centre of mass is the location of an object’s centre of mass. It is the average position of all parts of a system or mass distribution in space. A force is frequently applied at this moment, 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.

As a result of the forces exerted on the particles in a system by surrounding bodies or by the action of a force field, the motion at that point would be the same as the motion of a single particle, whose mass and motion are determined by the interaction of individual particles. The point where the force field acts is called the centre of mass of the particle system. It is useful to analyse the complex motion of a system of objects using the concept of centre of mass (COM), especially when two or more objects collide or when an object breaks into fragments.

How to Determine the centre of Mass

We may use the forces of gravity on a body to experimentally determine the centre of mass of a body. This is possible due to the fact that the centre of mass in a parallel gravitational field near the earth’s surface is the same as the centre of gravity. A body with an axis of symmetry and constant density will also have its centre of mass on this axis. Similarly, the centre of mass of a circular cylinder with constant density will be located on the cylinder’s axis. The COM of a body with constant density and spherical symmetry is at the centre of the sphere. If we consider it in a broad sense, the centre of mass of any body will primarily be a fixed point of that symmetry.

Centre of Gravity

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

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.

Consider a system of n particles with a total mass that is constant across time. Consider the system’s particle ‘i.’ Its position in the graph is denoted by the vector ri , and the centre of mass of the given system of particles is designated by the letter ‘R.’

This means that the motion of a system of particles’ centre of mass is unaffected by internal forces. The centre of mass moves as if the system’s entire mass is concentrated there, and an external force is operating on it.

The product of the total mass of the system and the velocity of the centre of mass equals the total momentum of the system of particles, which is given by:Fext =dp/dt. This is Newton’s second law of motion, which applies to a particle system.

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 major property of the centre of mass is that it appears to carry the entire mass of the body. The terms centre of mass and centre of gravity are frequently used interchangeably. The centre of mass is the average position of an object’s mass. There’s also the centre of gravity, which is where gravity appears to act.