Moment Of Inertia

The moment of inertia is a property of objects which can be referred to as ‘the inertia of a rotating body.’ This property helps the objects to withstand the angular acceleration by resisting it. Moreover, a moment of inertia can be calculated. The following article covers curated examples, formulas, influencing factors of the moment of inertia alongside the calculating methods.

What is the moment of inertia?

The moment of Inertia is the inertia of an item that is in rotation motion. It is also referred to as the mass moment of inertia, rotational inertia, second moment of mass, and angular mass. A moment of inertia is the quality of a body that resists a change in angular acceleration. Furthermore, it is affected by the rotational axis, mass dispersal, as well as other attributes. When there is a larger moment of inertia, more torque (force × distance from the axis of rotation) is required to change the angular acceleration. 

The axis of rotation is a fictitious axis upon which the movement of the items occurs. At the same time for calculating the moment of inertia (the formula we will study later in this topic only), the mass of an object is considered distributed equally on the centre of mass. The simple pendulum and the revolving car in stunts are some examples. The simple pendulum keeps its angular momentum continuing until external forces of gravity, friction, and other torques are applied.

Example of the moment of inertia:

Let’s assume you’re on a train. You grab a seat and take it. The train begins to move forward. After several minutes, the next station is coming, and the train starts slowing down, and lastly, it stops. At the very moment, you can observe that when the train came to a stop, your upper torso moved forward while your lower body remained motionless. This is due to inertia.

Affecting factors: 

The moment of inertia depends on several factors. Some of the factors, such as the structure and size of the item, are considered for calculations—the distribution of mass over the mass of the body. And lastly, the orientation of position length from the mass centre of the axis of rotation.

Moment of inertia formula: 

Moment of inertia, when calculated along the axis of rotation, equals the mass of the system multiplied by the square of the distance between the axis of rotation and centre of mass (a point where we assume all the weight to be concentrated). Its SI unit is a kilogram in product with a metre square. The moment of inertia formula is as follows.

I = Σ miri2

In the above formula, m denotes a product of mass, and r denotes the distance from the axis of rotation. 

Radius of gyration: 

The formula can express the rotational inertia of almost any element in an axial direction throughout its CG. In the formula, Mk2 = I, I depicts the moment of inertia, M mass, and k is the length of, say, distance from the axis. The k in the above formula is the radius of gyration. In a multidimensional body, every particle moves while carrying a different moment of inertia. As every point has a different distance from the axis of rotation, the radius of gyration is the summation of all those radii. 

For eg., k = √(R12 + R22 + R32…….)

How is a moment of inertia calculated?

The first step in calculating the moment of inertia for a body is to locate the corresponding axes. The precision of the calculations is highly dependent on how the sides are defined. Moreover, two theorems have been established to calculate the moment of inertia: The Parallel Axis Theorem and the Perpendicular Axis Theorem.

Parallel axis theorem:

The parallel axis theorem refers to the moment of inertia of an object in an axis parallel to the body, and passing through its centre is the overall moment of inertia of a rigid object. It is about the axial direction stationed at the centre and the product of the body’s mass times the square of a difference between two axes. The formula is as follows,

I = Ic + Mh2

In the formula, I depict the MOI of the body, and Ic denotes the MOI of the centre. M stands for mass and h2 for the square of the distance among both axes.

Perpendicular axis theorem

As the Perpendicular axis theorem states, the moment of inertia or rotational inertia of a planar item about an axis at a right angle to the surface is just the l amount of the rotational inertia of two axes passing through the same spot in the plane of the item. The formula is as follows

IZ = Ix + Iy 

I denote the MOI, and the subscripts are the denotations of axes, respectively. 

Conclusion:

From the above discussion, we can conclude that the moment of inertia is a moment that is caused by inertia on a rotating body. An example can be a movement that happens due to a sudden break in a train or bus. For deep understanding, we learned all general concepts of the moment of inertia and its formula, calculations, and constituents. The moment of inertia is a vector quantity which is calculated with the formula: I = Σ miri2. Moreover, the concept can be understood more deeply with parallel and perpendicular axis theorems, which describe the condition of the moment of inertia with an axis at different positions.