Moment of Inertia: Overview

Moment of inertia refers to the statistical measure of the rotational inertia of the object or the body. The moment is inertia however is the total of the products acquired by the accumulation of the mass of each part of the body by the quad of the gap from the axis. The S.I. unit of inertia is kg m2.

Moment of inertia= summation of mir2i   Moment of inertia can also be called the ratio of the total angular momentum angular velocity    (L) angular velocity (𝔀) under the principal axis. So this makes the formula as:

I= L / 𝔀

Moment of inertia takes place in both flat and non-linear movement. But for the flat movement, it has a single scalar and for non-linear movement,the calculations build up a 3*3 matrix of inertia moment which is called inertia matrix.

Factors on Which Moment of Inertia Depends

The factors on which moment of inertia depends are as follows:

1. The thickness of the material 
2. Measurement and form of the body 
3. The axis on which the body will move is in a circle. 

Moment of Inertia of Particles

The moment of inertia of the particles is known for the time when it completely relies on the mass and the gap from the rotational axis. Thus the formula for this is given below:

Moment of inertia = Mr2. 

In this M refers to the mass of the body and r is the gap of the tough body with consideration of the axis.

Moment of Inertia of Rigid Bodies

The rigid body is the sum-up of an unlimited number of particles. So we merge the different elements for the complete rigid body to get the valuation of the moment of inertia. The formula is given below:

Moment of inertia = summation of r2dm

Where r refers to the gap of the rigid body with consideration of the axis and dm refers to the differential mass element.

Some Real-Life Examples

Examples of the moment of inertia are:

• The earth is revolving in its axis with the old angular velocity, this is because of the moment of inertia 
• A rotating wheel is placed in toy cars so that when we rub the tires on the floor and leave the car then because of the moment of inertia, the car makes its way and moves forward for some instance.
• Every motor consists of a wheel attached which is huge and of high weight with its pressure on the edge, which leads to a high moment of inertia. The torque helps in driving the shaft to keep on expanding. So the revolution of the shaft may or may not be constant due to rotating wheel with high inertia, so the shaft moves with nearly consistent speed
• The wheel of the cycle continues to rotate even after stopping paddling because in the tire most of the mass focuses on the rims which create high inertia.

Application of Moment of Inertia

Few sums of the moment of inertia are:

• A body with a mass of about 600gm is revolving in its axis. The gap between the center of the mass from the body and the axis of rotation is 1.4m. Find out the moment of inertia of the body about the axis of rotation;

A. Here,

M= 600 gm = 0.6kg and r = 1.4m 

So, therefore, Moment of inertia = Mr2

Moment of inertia = 0.6*(1.4)2

= 1.176 kg m2.

• A body with a mass of 0.2 kg is rotating on its axis. If the gap between the center of the mass of the body and the axis of rotation is 0.8m, then find out the moment of inertia of the body.

A. Here,

M= 0.2 kg and r = 0.8m 

So according to the formula:

Moment of inertia = Mr2

= 0.2*(0.8)2

                              = 0.128 kg m2.

Conclusion

Moment of inertia is the calculation of the required force to help in the rotation of an object. The value can be changed to help in the reduction and increase of inertia. By increasing the radius from the axis of rotation, the moment of inertia increases and in turn slows down the speed of rotation. E.g.- if an athlete wants to increase the speed of rotation, then they must decrease the radius by bringing the segments of the body closer to the axis of rotation which decreases the radius and the moment of inertia.