Why is rotational inertia ∫r2dm?

What if you want to see this in a video?

In the rotational motion of a single particle, 

we are τ=Iα, rxF=mr2α

I know Newton's Second Law on the Rotation Movement.

What if you're curious about the rotational motion of a single particle?

Rotational inertia is the physical quantity associated with the rotational motion of a particle.

The greater the rotational inertia, the greater the tendency to maintain one's state of rotational motion.

So even if you apply force to rotate it, it doesn't rotate well.

It's like a mass object that doesn't want to accelerate well even if the force is applied.

This rotational inertia I has a value of mr2 in a single particle.

m would be the mass of a particle and r would mean the vertical distance from the axis of rotation to the particle.

But in fact, it's hard to see objects rotating in this particle form around us.

Consider a rigid body rotating around a fixed axis.

Where rigidity is an object that always has a constant distance between components.

It's hard, it's hard. Most of the objects around us are strong.

But technically speaking, there are few perfect rigid bodies, but let's say we're perfect rigid bodies.

We're going to split this rigid body into tiny pieces.

I'm not saying I'm really splitting it.

Let's split into tiny, virtually uniform volumes of steel in our heads, nearly zero pieces.

At that time, the mass of a piece of steel is very small, so it's called a micro-mass, which means a very small mass.

If we add the dm of every piece we share, it's the total mass of the rigid body.

There are external forces working on this little piece.

These external forces now create a spinning force on each piece.

But the rigid body we thought of was an object that rotated only on a fixed axis.

Therefore, only τz, the z-axis component of the turning force involved in the rotation of the turning axis during this turning force, is a significant force in the rotation of the rigid body.

The torquez then produces angular acceleration that rotates around the z-axis on each piece.

A rigid body must always have the same angular acceleration αz produced because the distances of all components are always the same.

Otherwise, the strong body will be crushed to pieces.

The rotational inertia of each piece is multiplied by the rotation axis and the square of the straight distance R from the piece and the mass dm.

So the relationship between zz and the angular acceleration of each piece is as follows.

We're going to add all the spinning forces that work on the rigid body to create a full rolling force.

This is called τnet.

The addition of the rotational inertia of all the pieces is the rotational inertia of the entire rigid body, which indicates the proportional relationship between the turning force and the angular acceleration when the z-axis directional force is applied to the rigid body.

This is called Iz because it is the rotational inertia of the rotation of the Z-axis.

Iz can get R2dm by integrating it with the entire rigid body. Because adding all the pieces together is like integral.

But there's one question.

When the spin force is applied to the actual rigid body, it acts on either part or on some part, and it does not work on any of these fragments.

Therefore, the proper amount of spinning force acting on the rigid body should be the sum of a few spinning forces, not the sum of many turning forces.

But torque is vector. Newton showed that vectors can be broken down by various forces.

So it doesn't matter how the spin force worked on the rigid body, how it did it, as we assumed above, how the spinning force worked on all the pieces of the rigid body, τ1, τ2, τ3, τ4, etc.

And it's a very reasonable idea when you think about how the force exerted on the external rigidity is applied inside the rigid body.

And then always turn the net into Iα.

Oh, of course, it's based on one fixed axis.

Let's take a look at Iz of rotational inertia.

Iz is the sum of the square of R and the mass dm of the vertical distance between the axis of rotation and the piece.

Rotational inertia increases with the larger the mass of the rigid body.

Also, the mass distribution of the rigid body grows further away from the axis of rotation.

You take an umbrella with you on a rainy day and turn it around in a good mood.

 Because the umbrella is wide open, the distribution of mass in the umbrella is far from the axis of rotation.

Then, because of the large amount of rotational inertia, it will put a lot of effort into your hands and the umbrella won't spin as fast as you think.

Let's close the umbrella and rotate it this time.

Because the mass distribution is closer to the axis of rotation, the rotational inertia becomes smaller.

You can rotate your umbrella more easily than before.

Of course, we'll be soaked in the rain, but it won't matter because we're in a good mood.