Centre of Mass of a Rigid Body

Centre of mass of a rigid body

Introduction

Let’s start by defining what a centre of mass is and how it is affected by various factors. A system’s centre of mass is defined as the point where all of the system’s mass appears to be concentrated. This point can be affected by an external force. The centre of mass of a rigid body is fixed and it will be located at the centroid (if the body has a uniform density). This article will help you learn about the centre of mass of a rigid body, its definition, equation and formula. 

Define centre of mass

The centre of mass is a position that is relative to an object. It is the mean location of mass distribution in space, or it is the average position of all the system parts. It’s a point where force is usually applied to produce linear acceleration rather than angular acceleration.

The centre of mass of a uniform disc shape, for example, would be at its centre. The object’s centre of mass does not always fall in the same place. The centre of mass of a ring, for example, is at its centre, where no material exists.

Two factors that determine the position of the centre of mass of a rigid body are:

• Body structure
• The mass distribution

These factors determine whether the centre of mass is located inside or outside the body. 

Inside the body, the body’s centre of mass is found in the physical structure (matter). But outside the body, the centre of mass is located on the body’s space and not on its physical structure (matter).

What purpose does the centre of mass serve?

The centre of mass of an object or system is that point where any uniform force on the object acts.This is useful because it simplifies the solution of mechanical problems involving describing the motion of oddly-shaped objects.

For calculation purposes, we can treat an oddly shaped object as if all of its mass is concentrated in a tiny thing at the centre of mass. This fictitious object is sometimes referred to as a point mass.

Gravity’s centre

Every differential element of mass experiences the gravitational force in a rigid body. The centre of gravity is when the rigid body’s total weight can be considered concentrated.

A body’s centre of gravity is a point where the resultant torque due to gravitational force is zero.

Since the value of gravitational acceleration remains the same for every element of the body, bodies with smaller dimensions, especially height, will have a centre of gravity and a centre of mass at the same point. However, if the body is of significant height, the variation of gravitational acceleration becomes substantial, then the centre of gravity and centre of mass will not coincide.

Centre of mass- point objects

(m1+m2) rcm =m1 r1+m2 r2 

Here, m1 and m2 pertains to mass of object one and two and rcm pertains to distance of the centre of mass from point of reference (origin). 

r1 is distance of object one from point of reference and r2 is  distance of object two from point of reference.

If we have point objects, we must use a different approach and formula that is listed above. If we need to find the centre of mass of an extended object, such as a rod, we must take a different approach. A differential mass and its position must be considered and integrated over the entire length.

Identifying the mass centre

We can use gravity forces on the body to determine the centre of mass of a rigid body in an experimental setting. This is possible because the centre of mass in the parallel gravity field near the earth’s surface is the same as the centre of gravity. Furthermore, a body with a symmetry axis and constant density will have its centre of mass on this axis.

Similarly, the centre of mass of a circular cylinder with constant density will be on the cylinder’s axis. The centre of mass will be at the sphere’s centre if we discuss a spherically symmetric body with constant density. If we consider it in a broad sense, the centre of mass of anybody will almost always be a fixed point of that symmetry.

The particle system and the centre of mass

We’ve only dealt with rigid body translational motion so far, where a rigid body is also treated as a particle. When a rigid body rotates, all of its constituent particles do not move in the same way. Nonetheless, we must treat it as a system of particles where all particles are rigidly connected.

On the contrary, we may have particles or bodies that are not rigidly connected but interact with one another via internal forces. Despite the complex motion that a particle system is capable of, the system’s translational motion is characterised by a single point known as the centre of mass (or) mass centre.

Conclusion

Particles have no dimensions and are only the size of a point. The distance between any two points on a rigid body remains constant at all times. The system’s entire mass can be considered concentrated at the centre of mass. The centre of mass for a system of two equal-mass particles is located at the intersection of the two particles’ lines.