Systems of particles

To understand systems of particles, we must first define a rigid body. 

The following are the features of a rigid body:

• The distance between particles in a rigid body remains constant despite the application of a constant force.
• A non-fixed rigid body can move in any direction or several directions at the same time.
• The angular velocity of a rotating rigid body is constant when there is no external torque.

Centre of Mass

The point at which the total mass of an object is concentrated is called the centre of mass. If someone applies force externally at this point, there is no change in its position in case the object is resting. 

The Motion of the Centre of Mass

The centre of mass of a particle moves as if the entire mass of the framework were accumulated at this place and all external forces were applied there. It will have constant energy if no external factor follows up on the body.

Torque

In rotational motion, torque is the ability to rotate an object about a fixed axis. Mathematically, it can be expressed as:

𝛕 = r x F 

Here,𝛕 is the torque, F is the force, and  r is the perpendicular distance.

The relationship between angular and linear velocity

Assume that a rigid body spins around any rotational axis with angular velocity (ω). If the perpendicular distance of particle ‘a’ from the fixed axis is ‘r’ and any particle in the rigid system has a linear velocity of ‘v’, then the relationship between them is expressed as

r x ω = v

The centre of gravity 

It is the location on the body where the entire weight of the body is meant to be focused. A body’s centre of gravity is defined as the point where the entire gravitational torque acting on it is zero.

Moment of Inertia

In rotational motion, the moment of inertia is the magnitude by which a rotating object opposes its motion. Therefore, it is also referred to as the rotational inertia of the object. 

Mathematically, it can be expressed as the product of the distance of the object from the axis and its mass. Rotational inertia is usually written as L in mathematical formulae and relations. 

The unit of rotational moment of inertia is ‘kg m2’. The moment of inertia of a rigid object rotating about its axis can be given by the below formula:

Here, m1, m2, and m3 are the masses of any three particles of the rigid object, and r1, r2, and r3 are their respective distances from the axis of rotation.

Radius of gyration 

An object’s radius of gyration is defined as the root mean square of the distance of the particles from its rotational axis. It is usually written as K in mathematical formulae and relations. It can be given by the following equation:

If we multiply an object’s mass with the radius of the gyration square, it is equal to the rotational inertia of the object. 

Therefore, 

I = MK2 

Conservation law for momentum

Momentum is defined as the product of mass and velocity. It depends on the force applied to the system of particles. 

Mathematically, if we consider two particles A and B, then momentum is written as:

A = m1 (vf1 – vi1) …. (i)

B = m2 (vf2 – vi2) …. (ii)

Combining equations (i) and (ii) and comparing them with Newton’s second law of motion, the conservation law of momentum can be expressed as:

m1u1 + m2u2 = m1v1 + m2v2 …… (iii)

Angular momentum of a system of particles

Based on the theories of rotational motion, we can determine the angular momentum of a body by L and can measure it from a fixed point where the vector momentums of the individual particles are concentrated.

Thus, 

L = L1 + L2 + L3 + L4 + … + Ln … (iv)

Kinetic and potential energy of a system of particles

When a body is at rest, it possesses potential energy denoted by U. In the state of motion, the body possesses kinetic energy denoted by K. The mathematical expressions are written as:

U = mgh

K = ½ (mv2)

where ‘m’ is the mass, ‘g’ is the acceleration due to gravity, ‘h’ is the height from sea level, and ‘v’ is the velocity with which the body is moving.

According to the law of conservation of energy, energy cannot be created or destroyed. When the body starts gaining velocity, the potential energy is converted to kinetic energy. This energy increases till the body attains a rest position, in which the kinetic energy is converted back into potential energy.

Conclusion 

This article describes the systems of particles and rotational motion. The theory of the systems of particles is used to determine the force and centre of mass of rigid bodies combining pulleys, multiple body blocks, and other factors. A rigid body’s rolling motion is a combination of rotational and translational motion. The total angular momentum of a rigid body or a system of particles is conserved if there is no external torque acting on it. This is in accordance with the law of conservation of angular momentum.