Moment of Inertia

# Moment of Inertia

When dealing with the motion of a rigid body, there are two forms of motion that it can have.

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A motion is translational if any straight line inside the body remains in the same direction during the motion and all of the particles that make up the body move in the same direction.

The motion is called rectilinear translation if these pathways are straight lines, such as an automobile traveling in a straight line or an object dropping vertically downhill.

The motion is a curvilinear translation if the pathways are curved lines, such as an object moving on a curved path under gravity (projectile motion).

When all of the particles describe circular routes around a line called the axis of rotation, the motion is rotational around that axis.

While translational motion may be easily described using existing principles, rotational motion can be quite difficult due to the fact that rotating motion cannot be one-dimensional by definition.

Rotational motion in a plane, or two-dimensional or plane rotational motion in which the body rotates around a fixed axis, is the most basic type of rotation. This is the type of rotational motion we’ll be talking about right now, also known as pure rotational motion.

Illustrative picture depicting an example of rotational motion, The problem becomes much more complicated if the axis is not fixed, thus we will not cover the general case of translation with rotation.

We need some new terms to characterize the particular properties of rotating motion. These new concepts, such as the moment of inertia, torque, and angular momentum, can be simply deduced from previously understood concepts.

## A brief outline:

However, in this article, we will be discussing only the concept of Moment of Inertia, Moment of Inertia formula, and Moment of Inertia for a rigid body and for a system of bodies.

## What is the Moment of Inertia or Rotational Inertia of the body?

Newton’s first law of motion states that a body cannot change its state of rest or uniform motion in a straight line by itself. The inertia of the body refers to its immobility or inability. To modify the state of rest or uniform motion in a straight line of the body, an external force is obviously necessary. The more power required to change a body’s state, the greater its inertia. Newton’s second law states that in translatory motion,

Mass multiplied by acceleration equals force (F=ma)

As a result, the force required to produce or destroy a given acceleration is directly proportional to the mass of the body. Obviously, the more mass a body has, the more force is required to alter its state, and hence the greater its inertia.

It’s also true in the other direction. As a result, the body’s mass is a measure of its inertia in translational motion.

In the case of rotatory motion, the property of inertia also applies. Unless a torque is supplied to the body, it will tend to maintain its state of rest or uniform rotation around a particular axis.

The property of a body that opposes the torque that tries to modify its state of rest or uniform rotation along a specific axis is known as the moment of inertia or rotational inertia of the body about that axis.

A body’s moment of inertia about a given axis plays the same role in rotational motion around that axis as its mass does in translational motion. The two, however, are not the same. Whereas a body’s inertia is only determined by its mass, the moment of inertia of a body around a certain axis is determined by:

1. Its mass,
2. The axis of rotation’s position and direction. Because of this, two bodies of the same mass and shape that rotate around distinct axes of rotation will have different moments of inertia.
3. Body morphology (i.e., distribution of the mass of the body about the said axis). Because of this, two bodies with the same mass but different forms will have different moments of inertia around the same rotational axis. A sphere and a disc of equal mass and dimension, for example, have different moments of inertia around the same axis of rotation.

Quantitatively, the moment of inertia of a body about a given axis is equal to the sum of the products of the masses of the constituent particles and squares of the distances perpendicular to the axis of rotation.

If m 1, m 2, m 3, m 4……. Are the masses of particles situated at a distance r 1, r 2, r 3, r 4 respectively from the axis of rotation, then the moment of Inertia is defined as

I=m 1 r 12+m 2 r 22+m 3 r 3 2+m 4 r 4 2

I=m r 2 (g c m 2)

= summation of the said products for all particles constituting the body.

whereas,

Moment of Inertia of rigid body is defined as,

I= ∫(d m)r 2

Where r is the distance of a mass element of mass d m taken from the axis of rotation.

### Theorems for Moment of Inertia:

1. Perpendicular axis theorem
2. Parallel axis theorem

If we know a body’s moment of inertia around one axis, we may apply the following two theorems to determine its moment of inertia around another axis.

## The theorem of Perpendicular Axes:

A lamina’s moment of inertia at an axis perpendicular to its plane is equal to the sum of its moments of inertia about any pair of mutually perpendicular axes in its own plane if the given axis passes through their point of intersection.

I z=I x+I y

### The theorem of Parallel Axes (Steiner’s Theorem):

A body’s moment of inertia about any axis is equal to its moment of inertia about a parallel axis through its center of mass plus the product of its mass and the square of the perpendicular distance of its center of mass from the given axis.

I=I cm+M h 2

h= Distance between the axes AB & CD

M= Mass of the body

### Bodies of regular shape have a moment of inertia:

 Body Moment of Inertia Axes Thin Rod 1 ⁄12 M l 2 Perpendicular to the rod at mid point Annular Disc 1⁄ 2 M(R 12+R 22) Through its center and perpendicular to its plane Annular Disc 1 ⁄2 M(R 12+R 22) Its diameter Solid Cylinder 1 ⁄2 M R 2 Axis of the cylinder Solid Cylinder M[R 2 ⁄4+l 2 ⁄12] Perpendicular to the axis of the cylinder and passing through its center of mass.

### A system of particles has a moment of inertia:

Inertia is defined as the ability of an object to maintain its current state of motion unless it is acted upon. In the case of linear motion, inertia is measured as mass, and in the case of angular motion as the moment of inertia. An object’s moment of inertia largely depends on the distribution of its mass within the body, in addition to its mass. It can therefore be possible for two bodies of the same mass to possess different moments of inertia.

If we consider a rigid body as a system of particles whose relative position does not change, we can say that the particles are rigid.

Question 1: In what ways does rotational motion come into play in everyday life?

Answer: Rotational motion is the motion of wheels, gears, motors, and other rotating objects. The rotatory motion of the helicopter blades is likewise rotatory motion. When you open or close a door, it swivels on its hinges. In an amusement park, a spinning top mimics the action of a Ferris Wheel.

Question 2: What is the mechanism by which rotation improves stability?

Answer: A symmetric body with no torques applied and even a small amount of internal damping (as all real objects do) will eventually rotate around its principal axis with the smallest moment of inertia. The more effort it takes to change the spin axis (= better stability), the faster the spin is (= higher angular momentum).

Question 3: How can rotational inertia be reduced?

Answer: Drilling and/or cutting bulk away from your wheel spokes is the best approach to reduce rotational inertia. By drilling and/or cutting on the outer rim of the wheel with a drill and/or Dremel tool, mass can be eliminated.

Question 4: What’s the difference between rotational and circular motion?

The object in a circular motion just moves in a circle. Artificial satellites, for example, orbit the Earth at a fixed altitude. The item rotates around an axis in rotational motion. We can consider any planet that rotates on its own axis.

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