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