Dimensions of Moment of Inertia

A body's moment of inertia is its capacity to resist angular acceleration. It calculates the amount of torque required by a body to achieve the desired angular acceleration on a rotating axis. The moment of inertia of a body is supplied by the total of the moments of inertia of each particle, and the moment of inertia of each particle is given by the product of the mass and the square of the particle's distance from the rotational axis.

The body's resistance to having its rotational speed along an axis changed by torque is measured in MOI. The axis may be internal or external, and it may or may not be fixed. The axis might be internal or exterior, and it can be fixed or not. The moment of inertia is comparable to mass in linear motion when calculating angular momentum for a rigid body. A composite unit of measurement is the moment of inertia unit. In the United States, m is measured in slugs (1 slug = 32.2 pounds) and r is measured in feet, with I represented in units of the slug-foot square.

Importance of Moment of Inertia:

The moment of inertia is an essential subject that is addressed in the majority of physics problems that involve mass in rotating motion. MOI is commonly used to compute angular momentum. In the next paragraphs, we will learn more about this subject.

The moment of inertia is defined as the amount indicated by the body resisting angular acceleration, which is the sum of the product of each particle's mass and its square of the distance from the axis of rotation. In simpler terms, it is a number that determines the amount of torque required for a certain angular acceleration in a rotating axis. The angular mass or rotational inertia are other names for the moment of inertia. 

The moment of inertia is frequently expressed in relation to a certain axis of rotation. MOI changes based on the axis used.

In general, Moment of Inertia is written as  I = m × r²

where,

m = The sum of the mass products.

r = Distance from the rotation's axis.

as well as the integral form: I = ∫dI = ∫0M r² dm

M¹ L² T⁰ is the dimensional formula for the moment of inertia.

In linear motion, the moment of inertia plays the same role as mass. It is the measurement of a body's resistance to a change in rotational motion. It is constant for a specified rigid frame and axis of rotation.

MOI, I = I = ∑mi ri². . . . . . . (1)

KE, K = K = ½ I ω² . . . . . . . . . (2)

The Moment of Inertia is affected by the Following Factors:

The material's density

The body's shape and size

Rotational axis

Rotating body systems are further classified as follows:

Separate (System of particles)

Consistent (Rigid body)

The integration approach is used to calculate the moment of inertia of continuous mass distribution. If the system is split into an infinitesimal mass element 'dm,' and 'x' is the distance between the mass element and the axis of rotation, the moment of inertia is:

I = ∫ r2 dm . . . . . . (3)

As seen in the table above, the moment of inertia is affected by the axis of rotation. Whatever we've determined up to this point is the moment of inertia of those objects when the axis passes through their centre of mass. After selecting two distinct axes, you will see that the item resists rotational change differently. 

The Following Theorems may be used to calculate the Moment of Inertia along Any given Axis:

The theorem of Parallel Axis

The theorem of the Perpendicular Axis

The moment of inertia around a parallel axis through the centre of mass is given by,

I = Icm + Md²

If the moment of inertia (I) of a body of mass m about an axis is expressed as:

I = Mk²

In this case, k is referred to as the radius of gyration of the body about the specified axis. It is the radial distance from the specified axis of rotation at which the full mass of the body can be considered to be concentrated for its rotational inertia to stay constant.

A solid sphere's radius of gyration about its axis is:

Mk \[^{2}\] = 2 \[\frac{MR^{2}}{5}\] , or k = \[\sqrt{\frac{2}{5}}\] \[\times\] R.