Moment of Inertia

Introduction

The moment of inertia (MOI) is a crucial topic in physics involving rotational mass. This is a specific quality, deciding the torque required to attain the particular angular acceleration in the rotational axis. This is used to determine the angular momentum. The SI unit of MOI is kg/m2.

The MOI is not fixed but is dependent on the position and the orientation of the axis of rotation concerning a particular body. Several machines like automobile engines and steam engines produce rotational motion with a disc having a high moment of inertia, known as the flywheel. Because of the high MOI, the flywheel obstructs the sudden increase or decrease of speed of the particular vehicle. It provides a gradual change in the speed and smooth running of the vehicle. This is a typical example of MOI.

Moment of inertia

The moment of inertia is the quantitative measurement of the total rotational inertia of the body. The MOI is selected in line with the axis of rotation. The axis linked to the rotation may be external or internal and may not be fixed.

When the body’s mass obstructs the change in linear motion, it measures inertia in linear motion. When the moment of inertia obstructs the change in the rotational motion, it is termed as the measure of inertia in rotational motion.

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 is 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 changes the entire MOI of the body.

The rotating body can be divided into two parts.

• The system of particles (Discrete)
• The Rigid body (Continuous)

The formula for moment of inertia

After understanding the moment of inertia, let us look at the inertia formula in detail.

The general MOI is mathematically defined as

I =  mr²

m = Total mass of the product

r = Total distance from the axis of rotation

The dimensional moment of inertia formula is defined as

I = M¹L²T⁰

Considering the formula, the role of the mass in a linear motion is the same as the moment of inertia.

The moment of inertia for a system of particles and rigid particles

Moment of inertia of a system of particles

The below-mentioned formula mathematically calculates the moment of inertia of a system of particles.

I = ∑miri²

Moment of inertia of rigid bodies

The moment of inertia of rigid bodies is mathematically defined through the integration method. For finding out the moment of inertia of rigid bodies, the system should be divided into an infinite number of small masses, ‘dm,’ and ‘r’ is the object’s distance from the axis of rotation.

I =  ∫ r²dm

Using the moment as mentioned earlier of the inertia formula for the system of particles and rigid bodies, you can easily find out the MOI of the required object.

Parallel axis theorem

The parallel axis theorem is an important theorem discussing the moment of inertia. The theorem states that the total inertia of a body is the total sum of the MOI of the body about the parallel axis and the product of mass and the square of the distance between the two parallel axes.

Iz = Iz’ + Md²

Iz and Iz’ = Moment of inertia about the z and the z’ axis

M – Total mass of the body

d = Total perpendicular distance in between two parallel axes

This theorem can be applied to any kind of body of any shape. This theorem finds the inertia of the body from any axis.

Radius of gyration

The radius of gyration represents the moment of inertia around an axis. The radius of gyration can be defined as the imaginary distance from the centroid. The cross-section area is imagined to be focused from the point to attain the same inertia.

The moment of inertia formula concerning the radius of gyration can be mathematically defined as

I = mk² ———— (1)

From (1), generate the radius of gyration.

k = √I/m

M = Total mass of the body

k = Radius of gyration

The SI unit of k is “mm”. With the help of k, it gets easy to find the MOI of various complicated objects.

A body with n number of particles whose axis of rotation has r₁, r₂, r₃, …, r_n.

I = m₁r₁ + m₂r₂ + m₃r₃ + …. + m_nr_n

Considering the mass of all the particles is the same.

I = m(r₁ + r₂ + r₃ +…+ r_n)

I = mn(r₁ + r₂ + r₃ +…+ r_n)/n

Equating mn as M signifies the total mass of the body

I = M[(r1 + r2 + r3 +…+ rn)/n]

Mk² = M[(r1 + r2 + r3 +…+ rn)/n]

k = √(r1 + r2 + r3 +…+ rn)/n

The Radius of gyration has various uses. Let us look at some uses of it.

• It compares how the various structural shapes would behave under compression along the axis
• It is also used to know the buckling in a compression beam

Conclusion

The moment of inertia has a vital role in determining the torque required to attain the specific angular acceleration in the rotational axis. in this article, we discussed the moment of inertia formula and the impact of moi on the system of particles and rigid bodies, along with the parallel axis theorem and radius of gyration. The moment of inertia is a critical topic in physics that plays an essential role in several entrance examinations.