Unit of Moment of Inertia

Introduction

The product of a body's mass and its squared distance from its axis of rotation determines its moment of inertia. Using the center of mass (COM) as an example, the moment of inertia is a concept developed from the concept of COM. A centre of mass represents the imaginary point in a body where all the mass of the body is concentrated. When a body is inertial, it is incapable of changing its position or uniformity of motion on its own. Because it takes more energy to change the state of an object with a higher mass, it has a higher inertia.

Moment of Inertia

It is the magnitude of resistance an object has, to do rotational changes. It needs to be defined in terms of a distinct axis of rotation. The M. O. I. about a given axis of rotation is a measure of the body's rotational inertia. It quantifies how various parts of the body are divided up at different intervals from the axis.

The M. O. I. of a rigid body is determined by

• Body mass index., shape, size.

• Body dimensions and shape.

• The distribution of mass around the rotational axes.

Any entity has a centroid, or center of gravity, which is the point within the object where gravity appears to act. An object at rest will remain at rest if its center of gravity is along a vertical line passing through it. A 2D surface's centroid corresponds to the center of gravity of the area.

Perpendicular axis Theorem

In plane areas, moments of inertia (M. O. I.) are equal to the sum of moments of inertia around mutually perpendicular axes.

That is to say, M. O. I. is

$$\mathrm{I_Z=I_X+I_Y}$$

Parallel axis Theorem

Moment of inertia about an axis parallel to the centroidal axis is summed over M. O. I. about the centroidal axis and mass of the body times the square of the distance from the reference axis.

$$\mathrm{I= I_{CM}+Mr^2}$$

Here, The moment of inertia about an axis parallel to the centroidal axis = I

The moment of inertia about the centroidal axis = $\mathrm{I_{CM}}$

The perpendicular distance between two centroidal axis = $\mathrm{r}$

By multiplying the force by the perpendicular distance between a point and the line of action, we can calculate the moment of a force applied to a point.

In every body there exists a point where the whole mass of a body can be assumed to be concentrated. The radial distance of this mass point and the axis of rotation is known as the radius of gyration (K). The moment of inertia can be calculated in terms of radius of gyration using the following formula.

$$\mathrm{I=MK^2}$$

Here, The Moment of inertial = $\mathrm{I}$

Mass of the body = $\mathrm{M}$

The radius of gyration = $\mathrm{K}$

Moment of Inertia of Rigid Bodies

Inertia

When a body is inertial, it is incapable of changing its position or uniformity of motion on its own. An object with a greater mass has a greater inertia, since it requires more force to change its state. A body's inertia is calculated based on its mass.

Rotational Inertia

Rotational inertia is the property of bodies that can rotate around an axis. An object's rotational inertia increases with the torque or couple needed to change its state of rotation. The rotational inertia of the body is expressed as the M. O. I. of the body about the rotating axis. With the change in the axis of rotation the rotational inertial also changes.

Factors affecting the moment of Inertia

The product of a body's mass and the squared of the distance from its axis of rotation determines its moment of inertia. The M. O. I. is a concept that evolved from the idea of a centre of mass. Moment of inertia refers to how strongly an object resists rotational changes. It must be defined in terms of a specific rotational axis. The rotational inertia of a body is measured by the M. O. I. about a specific axis of rotation. It measures how fractionally a body is divided into different parts at various distances from the axis.

The equation for a particle's moment of inertia is

$$\mathrm{I=mr^2}$$

A body's moment of inertia is determined by the following factors −

• Body mass index.

• Body dimensions and shape

• The distribution of mass around the rotational axes

• The alignment and placement of the rotational axes with respect to the body.

Unit of moment of Inertia

The equation for a body's M. O. I. is

$$\mathrm{I=mr^2}$$

The unit of mass is kg, the unit of length is metre.

The moment of inertia unit will be $\mathrm{kg.m^2}$.

Conclusion

The moment of inertia is a concept that evolved from the concept of a centre of mass. A centre of mass is an imaginary point in a body where all of the body's mass is concentrated. The product of a body's mass and its squared distance from its axis of rotation determines its moment of inertia.

When a body has moment of inertia, it is incapable of changing its position or uniformity of rotational motion on its own. An object with a greater mass has a greater moment of inertia, since it requires more moment of force to change its state of rotation. The unit of M. O. I. is $\mathrm{kg.m^2}$. The position of axis of rotation and distribution of mass determines the M. O. I. of a rigid particles.

FAQs

Q1. Is the moment of inertia affected by the locus of application of force?

Ans. No. The magnitude, direction, and locus of application of force determine torque. The moment of inertia is affected by mass as well as its distribution in relation to the rotational axis

Q2. Why does the moment of inertia remain constant?

Ans. No, Summation of the M. O. I. of the body's mass elements results in the body's total moment of inertia. In contrast to mass, which is constant for any given body, the M. O. I. is affected by the location of the centre of rotation. The M. O. I. is typically calculated using integral calculus.

Q3. Is inertia present in all objects?

Ans. As a force, inertia prevents stationary objects from moving and keeps moving objects at a constant velocity, it is a force that prevents all objects from stopping. Inertia exists in all objects. The inertia of a massive object is greater than that of a fairly small object.

Q4. What happens if the moment of inertia increases?

Ans. The greater the moment of inertia of an object, the more force will be needed to alter its rotational state. An object's resistance to changes in its state of rotation increases with increase in moment of inertia.