System of particles

A system of particles is described as a group of inter-related particles. Ideally, a rigid body comprises a system of particles with a well-defined shape. A rigid body or object has a perfect or close to a perfect shape. It also means that the object’s shape, size, or other physical features do not change in normal circumstances. Thus, the particles have a constant distance between them. However, no such body can exist, which means that no perfectly rigid body exists in nature. But in many situations, the deformations due to the movement of particles are negligible.

Centre of mass

In a system of particles, the centre of mass is the point where the total mass in translation motion is believed to be concentrated.

If all the forces acting on a system of particles or body were to be applied at its centre of mass, the body should remain unaffected. It means whether the body is in rest or motion, its position will remain unaffected. 

The centre of mass of a body is the balancing point of the system of particles or a particular body.

Note: If two particles have the same mass, then the centre of the mass lies at the midpoint of the line, joining them.

For a system of n particles of masses m1, m2, m3,…mn and their respective position vectors, the position is:

rcm = m1r1 + m2 r2m1 + m2

For a system of n particles of masses m1, m2, m3,…mn and their respective position vectors, then the position is:

rcm = m1r1 + m2r2+. . . . . .  + mnrnm1 + m2 + . . . .. + mn = i=1n mi rii=1n mi

Translation motion

When a body’s particles move together at a given instant of time, the velocity of all its particles is the same; it is called translation motion. 

Examples of translation motion can be vehicles moving in a straight line or a walking animal.

The motion of the centre of mass

The centre of mass of a system of particles moves where all the mass of the system of particles is concentrated at the centre of mass. And also, all the external forces applied are focused on this point. The velocity of the centre of mass of a system of two particles, m1 and m2, with velocities v1 and v2, is given by:

vcm =m1v1 + m2 v2m1 + m2

Acceleration of centre of mass acm of two systems of the particle is given by,

acm =m1a1 + m2 a2m1 + m2

In a condition where no external force is working on the body, the body’s centre of mass will have a constant momentum. Its velocity will remain constant, and acceleration will be zero, Mvcm = Constant.

Angular momentum

In various systems of particles, angular momentum is defined as the rotational equivalent of linear momentum. In simple terms, angular momentum is the product of angular mass or moment of inertia and angular velocity. The angular momentum about an axis of rotation is a vector quantity.

Angular momentum is given by,L=rp

SI unit of angular momentum is kgm2s-1Axis of rotation

A system of particles or a rigid body is said to be rotating if all the particles of the body move with the same angular speed but make circles of different radii. The centre of all the circles is the same. If we draw a line that passes through this common centre, this will be the axis of rotation.

A rigid body is considered in equilibrium if, under the action of forces, the said body remains at rest or in a uniform motion. If the vector sum of all the forces acting on a body is zero, the body is said to be in translational equilibrium. 

If the vector sum of torques of all the forces acting on that body about the reference point is zero, the body is said to be in rotational equilibrium. However, if both these conditions are satisfied, the body has incomplete equilibrium.

Moment of inertia

Moment of inertia is described as similar to the rotational inertia of a system of particles. The moment of body inertia about an axis is the sum of the products of masses and the square of their respective perpendicular distance from the axis. It is given by,

=dLdt

Moment of inertia

Moment of inertia is described as similar to the rotational inertia of a system of particles. The sum of the product of mass and the square of perpendicular distance give the moment of inertia. It is given by:

I = m1r12 + m2r22 + m3r32 +. . . . . . + mnrn2 = i=1nmir12

Conclusion

Every rigid or solid body is made up of a system of particles. In this system, the particles are interrelated and form a rigid body. However, in nature, no perfectly rigid body can exist, and the factors that come up during the movement of such bodies include the centre of mass, angular momentum, torque, and moment of inertia. The centre of mass of a body is also considered the balancing point of the body. It is a point in the body where the body’s entire mass is imagined to be concentrated. The angular momentum of a body and torque are interrelated.