PERPENDICULAR AXES THEOREM

INTRODUCTION-

The M.O.I of a 2-dimensional object about the axis that passes directly from it is equal to the sum of the M.O.I object about 2 adjacent axes lying on the object plane. According to the above definition the Perpendicular axis theorem can be labeled,

              IZZ = IXX + IYY.

The theory is that the inertia of a plane’s body in the axis perpendicular to its plane is equal to the number of inertia times in relation to the two surrounding axes lying on the body plane so that all three axes are connected and aligned. have a common point.

THEORY

To understand what a perpendicular axis theorem is, let’s consider something like a ball or a rotating disk that rotates around its center. You already know the time of inertia of an object about its center. But, After you change the point around this thing, how do you get the Moment of Inertia? To understand this, we need to know about the Perpendicular axis theorem.

Now, before we discuss the Perpendicular axis theorem, we will first see what the inertia period is.

Moment for Inertia

Angular acceleration resistance is defined as the time of an object’s motion. It is written as the sum of the products of the weight of each particle within the object, and the square of its distance from the rotating axis.

Let us consider something with a mass m.

Contains small particles with a mass of m1, m2, m3 …….. respectively.

The perpendicular distance of each particle from the center of gravity is r1, r2, r3 …… According to the definition of Moment of Inertia, the great moment of Inertia of everything is:

I  = m1r12 + m1r22 + m1r32 + …..

As we look at body weight (m) to be fixed somewhere. That point is its center of plurality. If this weight m is at a distance of r from the center of the weight it means that the Moment of Inertia of everything says, I = ∑ Mr^ 2

So to calculate the time of Inertia, we use two important theorems. The first is the Parallel Axis theorem and the second is the Perpendicular Axis theorem. In this article, we will only emphasize the perpendicular axis theorem. Let’s understand what this concept is all about.

Perpendicular Axis Theorem

Let us assume that there are three parallel axis called X, Y and Z. They met from O.

Now imagine an object lying on an XY plane with a small dA. It has a distance of y from the X-axis and a distance of x from the Y-axis. Its distance from its origin is r.

 Let IZ, IX and IY be Inertia times with the X, Y and Z axis respectively.

Inertia time with Z-axis i.e.

Iz = ∫ r2.dA …………. (i)

Here, r2 = x2 + y2

Enter this number in the number above

Izz = ∫ (x2 + y2). dA

Izz = ∫ (x2. dA + y2. dA)

Izz = Ixx + Iyy

Parallel  Axis  Theorem

Moment of Inertia about any axis is equal to the sum of the Moments of Inertia about an axis parallel to this axis, passing through the Centre Of Mass (COM) and the product of the mass of the body with the square of the perpendicular distance between the axis in consideration and the COM axis parallel to it. The corresponding axis theory states that if the body is rotated instead of a new z ′ axis, corresponding to the original axis and subtracted from it by distance d, then the moment of inertia I in relation to the new axis relative to -Icm by.

I0 = Ic + md2

Here,

I0 = M.O.I shape about point O

Ic = M.O.I shape about centroid (C)

md2 = Added M.O.I due to the distance between O and C

Use of the Perpendicular Axis Theorem 

If the amount of Inertia time about these two axes is known, then we can easily calculate the Moment of Inertia about the third axis using this theory.

By counting M.O.I. of 3-dimensional objects such as cylinders, we can use this theory. By doing this, break the cylinder into organizer disks and merge all M.O.I. compact discs.

CONCLUSION

Local inertia times about the centroid of common categories are found in standard tables. But most of the time you will come to a situation where you may need to calculate the moment of inertia for another axis that is not exceeding the centroid. The parallel axis theorem and the perpendicular axis theorem are useful in calculating the local moment of inertia of such conditions.