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Moments of Inertia Using Integral Method
Rebiaz Studio
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8/29/2024
Integral analysis is one of the methods of calculus. This time we will use the concept of integral to calculate the moment of inertia of a thin stick rotating on a certain axis.
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00:00
Hi colleagues, welcome to the Diggus Science channel.
00:05
Channel that presents animation videos related to educational material.
00:10
Previously, we could calculate the moment of inertia of the thin stick using the Riemann sum.
00:17
This method seems to be the method that logic can understand best.
00:22
There are other methods besides that.
00:27
One of them is the integral method.
00:30
What's that? Let's discuss.
00:36
We want to calculate the moment of inertia of the thin stick, but this time the shaft is at a quarter of its length.
00:48
Before we go any further, we need to place this thin stick in a three-dimensional Cartesian coordinate system.
00:55
More precisely on the x-axis.
00:59
And the axis of rotation passes through the coordinate center.
01:06
If the length of this stick is L, then one of the endpoints will be at coordinates minus a quarter L,
01:12
and the other end will be at coordinates 3 over 4 L.
01:18
In the integration process we must know the elements of integration.
01:22
The smallest element that makes up that object.
01:28
Suppose there is a very small element. Its length is dx and its mass is dm.
01:37
If this element is shifted from one endpoint towards the other endpoint, a stick is formed.
01:51
From here, you already understand what is meant by the elements of integration.
01:55
Elements that can form objects.
02:01
Using the integration method, the moment of inertia of a rigid body about a certain axis can be calculated through the equation I,
02:07
which is equal to the integral of r squared over the mass element dm.
02:14
What is meant by r here is the distance from the mass element to the axis.
02:21
Because it is on the x-axis, the distance is x.
02:26
r is the same as x.
02:31
Then what about the mass element dm?
02:36
Let's just say this stick is a homogeneous stick.
02:39
A stick that has uniform density at all points.
02:45
Therefore, dm per dx equals big M over L,
02:51
or dm equals big M over L dx.
02:58
We can substitute this dm value into the integration equation.
03:05
The next issue is the limits of integration.
03:09
To form a stick, the mass element moves from x equal to minus a quarter L to x equal to 3 over 4 L.
03:19
Thus the limit of integration is from minus a quarter L to 3 over 4 L.
03:26
The mass of the stick is fixed, and so is its length.
03:30
The big M value per L can be excluded from integral notation.
03:37
Now, the integration of x squared with respect to x is one third of x cubed.
03:45
Now, we can enter integration limits.
03:50
The value in brackets is 7 over 48 cubic L.
03:57
Thus, the value of the moment of inertia is 7 over 48 big ML squared.
04:07
This is the moment of inertia of a thin stick rotating about an axis passing through one quarter of its length.
04:14
We have understood certain integral concepts in the integral course.
04:18
This is not a complicated calculation.
04:23
Hope it is useful.
04:25
See you in the next video.
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