Systems of Particles and Rotational Motion

Rigid Body

In order to understand the rotational particle system, you must first define a rigid body. It is defined as:

• 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 in several directions at the same time.
• A rotating rigid body’s angular velocity is constant at all times.

What exactly is the Centre of Mass?

The point at which the total mass of the structure is designed to be focused for translational movement is known as the Centre of Mass. A two-molecule framework’s centre of mass is always on the line connecting the two particles and is positioned in the middle of the particles.

The motion of the Centre of Mass

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

Vector Product of two Vectors

ab  denotes the vector or cross result of two vectors a and b. The magnitude of such a vector is ab sin, and the vector’s direction is calculated using the right-hand rule method.

Cases of Mechanical Equilibrium in Rigid Bodies

The mechanical equilibrium of a rigid body can be determined by two situations in the chapter on systems of particles and rotational motion, such as:

• If Translational equilibrium = 0 and 
• If Rotational equilibrium = 0 

Then total external torque is 0 and where torque is a moment of force calculated by the magnitude of the force being multiplied and applied to the particle by the perpendicular distance of force from the particle’s axis of rotation.

The Relationship Between Angular and Linear Velocity

Assume that every rigid body spins around any rotational axis with angular velocity (ω). If particle a’s perpendicular distance 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 given by

ωr = v

Torque

Torque is the inclination of a force to spin the body to which it is applied. The torque is calculated by multiplying the magnitude of the component of the force vector located in the plane perpendicular to the axis by the shortest distance between the axis and the component. the force’s movement

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.

Assume that we have a system of n particles with masses W1, W2, W3…Wn travelling in a straight line at x1, x2, x3…xn distances from the origin. The system’s centre of gravity is then given by

Xcd = W1x1 + W2x2 + … Wnxn / W1 + W2 + … Wn

Moment Of Inertia

When an object spins around any axis, it has a tendency to resist its own motion, which is known as the moment of inertia. The letter I stands for it, and the SI unit is kg/m2. The product of an object’s mass and the square of its perpendicular distance to the rotating axis is its moment of inertia. I = MR²

System of Particles and Rotational Motion Theorems

The System of Particles and Rotational Motion is made up of two key theorems that must be grasped in order to fully comprehend the subject. The following are some theorems:

The Perpendicular Axis Theorem

The moment of inertia I of a body for a particular perpendicular axis is always equal to the sum of moments of inertia of that body about two axes in the plane of that body that are always at a 90° angle to each other and meet at a location where the perpendicular axis passes through, I = Ix+Iy.

The Parallel Axis Theorem

The Parallel Axis Theorem states that a body’s moment of inertia I about any axis is always equal to its moment of inertia Icm about a parallel axis.

I = Icm+ Ma2

Angular Momentum and Law of Conservation Angular Momentum

The product of momentum and the perpendicular distance of the line of momentum from the axis of rotation determines the magnitude of angular momentum around an axis of rotation, and its direction is perpendicular to the plane containing the momentum. The total angular momentum of a rigid body or a system of particles is conserved if no external force operates, according to the law of conservation of angular momentum.

The Rolling Motion

A rolling motion combines the translational and rotational motions of a spherical mass on a surface. When a body is moved, each particle has two velocities: one due to rotation and the other due to translation, with the vector sum of all velocities at all particles being the result.

Pure rolling and rolling with sliding are the two types of rolling motion. Pure rolling happens when there is no relative motion between the rolling body and the surface at the point of contact, and the body is considered to spin around this point of contact frame.

Conclusion 

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 pair acting, according to the law of conservation of angular momentum. Force, energy, and power are all related to rotational motion.