The moment of inertia

Moment of inertia is a particular topic utilized in various physics fields. It is commonly used in problems related to mass in rotational motion and for calculating angular momentum. It plays a significant role in rotational kinematics and linear kinetics.  The moment of inertia is also used to find out the kinetic energy, momentum, and Newton’s laws of motion for a rigid body. Rotational motion depends on the distribution of mass around the axis of rotation, which changes by changing the axis. Although, in rigid bodies, a moment of inertia is the summation of small quantities of mass multiplied by the square of the distance (distance from the axis).

Body: 

The moment of inertia of an extended rigid body is just the sum of all the little pieces of mass multiplied by the square of their distances from the rotating axis. For a body with regular shape and homogenous density, they are calculated using dimension, shape, and total mass.

Definition of Moment of inertia:

The moment of inertia is the resistance to the change in angular acceleration of the body. It is the sum of the product of the mass of every particle of the body and the square of the distance at which the object is placed from the rotating axis.

In simpler words, it can be said it’s the quantity that decides the amount of torque required for acceleration.

Moment of Inertia Formula 

The moment of inertia is the value due to the resisting angular acceleration and is the summation of the product of the mass of each particle with the distance squared. 

So, in simple words, 

Moment of inertia, I = Σm × r2 

where

mass = m

Value of distance from the rotating axis = r

On integration, 

I = ∫dI = ∫ r2 dm

And the dimensional formula for the moment of inertia can be described as an M1 L2 T0

The mass of inertia plays the same role as the mass of linear motion. It can also be measured as the body’s resistance by changing its rotational motion. Moment of inertia remains constant for rigid frame and rotation in specific axes. 

Moment of inertia, I = ∑mi ri2

Moment of Inertia of a system of particles

The Moment of inertia of a system of particles can be described as a,

I = ∑ mi ri2

ri = ith particle perpendicular distance from the axis. 

mi = Mass of  ith the particle 

Moment of inertia of rigid bodies 

The moment of inertia of rigid bodies can be calculated by integrating. If the system of rigid bodies is divided into an infinite number of particles, then its mass, ‘dm’ and distance of mass from the axis of rotation is ‘r’. Now, the moment of inertia becomes, 

I = ∫ r2 dm

Moment of Inertia of different Rigid bodies

Factors affecting the moment of inertia

Certain factors affect the MOI. Let us look at the elements in detail.

• The Shape and size of the body- The MOI is dependent on the size and shape of the body. The larger the mass of the body, the higher the inertia. 

• The axis of rotation- The inertia is dependent on the body’s axis of rotation. The inertia increases or decreases as per the increase or decrease of the radius of the axis of rotation.

• The density of the material- The density of the material plays a crucial part in generating inertia in an object. The inertia increases along with the increase in the density of the object.

The factors are significant as they determine the MOI for a body. The increase and decrease of a single factor change the entire MOI of the body.

Conclusion 

For a rigid body, the moment of inertia emerges as a physical parameter that integrates its shape and mass in momentum, kinetic energy, and Newton’s equations of motion.

It is applied in both linear and angular moments, but the appearance of the moment of inertia in planar and spatial movement differs significantly.

The moment of inertia in planar movement is defined by a single scalar quantity, however, in spatial movement, the same computations are given by a 3 * 3 matrix dimension of a moment of inertia.

The article highlights the definition of the moment of inertia, factors affecting it, the moment of inertia of a system of particles, and the Moment of inertia of rigid bodies.