Theorem of Perpendicular Axis with Step by Step Proof for JEE Main 2025
The parallel and perpendicular axis theorem deals with the moment of inertia. So, before studying the theorems, let's know about the moment of inertia, which is also important for JEE Main Exam. It is the property of a body by virtue of which a body resists angular acceleration.
Angular acceleration is the sum of the product of masses of particles of the body with the square of the distance from the axis of rotation.
Moment of inertia \[I_{i} = \sum m_{i}r_{i}^{2}\]
The equation is applicable only in-plane lamina.
This theorem is used for symmetric objects, i.e. when two out of three axes are symmetric. The moment of inertia of the third axis can be calculated by this equation when the moment of inertia of the other two axes is known.
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State and Prove Theorem of Perpendicular Axis
If you are wondering What Is The Theorem Of Perpendicular Axis, here is explanation for it: Perpendicular axis theorem statement - The perpendicular axis theorem states that the moment of inertia, for any axis which is perpendicular to the plane, is equal to the sum of any two perpendicular axes of the body which intersects with the first axis.
Let us consider a plane lamina made up of a large number of particles in the x-y plane as represented by the figure. Consider a particle having mass 'm' at point P.
From P, draw PN and PN' perpendicular to the x and y-axis, respectively.
The moment of inertia about the x-axis = my².
The moment of inertia of the whole lamina about the x-axis is given by
Ix = ∑my²-----(1)
The moment of inertia of the whole lamina about why the axis is given by
Iy = ∑mx²-------(2)
Similarly, the moment of inertia of the whole lamina about the z-axis is given by,
Iz = ∑mr²
But r² = x² + y²
Therefore,
Iz = ∑m (x² + y²)
From eq(1) and (2), we get:
i.e., Iz = ∑mx² + ∑my²
(or)
Iz = Ix + Iy.
The perpendicular axis theorem helps calculate the moment of inertia of a body where it's difficult to access one vital axis of the body.
Parallel Axis Theorem and Perpendicular Axis Theorem
The parallel and perpendicular axis theorem is mentioned below:
Parallel Axis Theorem
The parallel axis theorem states that the moment of inertia of a body about an axis that is parallel to the axis of the body and passing through its centre is equal to the sum of moments of inertia of the body about the axis passing through the centre of the product of the mass of the body and the square of the distance between two axes.
Statement
The parallel axis theorem is represented by the following equation:
I = Ic + Mh2
Where,
I = moment of inertia of the body.
Ic = moment of inertia about the centre of the body.
M = mass of the body.
h2 = square of the distance between the two axes.
Perpendicular Axis Theorem
Let’s see an example for this theorem:
Let us suppose that we want to calculate the moment of inertia of a uniform ring about its diameter.
Let the centre of the ring be MR2/2,
Where,
M = mass of the ring
R = radius of the ring
According to the perpendicular axis theorem,
Iz = Ix + Iy
Since it is a uniform ring; so all its diameter is equal
Therefore, \[I_{x} = I_{y}\] \[I_{z} = 2 I_{x}\]
Ix = Iy
∴ Iz = 2 Ix
Iz = (MR²)/2
Hence, the moment of inertia of a ring about its diameter is MR2/2.
Parallel Axis Theorem Derivation
Let Ic be the moment of inertia of an object with the axis passing through the centre of mass of the object. Let, I will be the moment of inertia about the axis A'B' (from figure AB) at a distance of h.
Let us consider a particle having mass ‘m’ and located at a distance ‘r’ from the centre of gravity of the body.
Distance from A’B’ = r + h
I = ∑m (r + h)2
I = ∑m (r2 + h2 + 2rh)
I = ∑mr2 + ∑mh2 + ∑2rh
I = Ic + h2∑m + 2h∑mr
I = Ic + Mh2 + 0
I = Ic + Mh2
Difference Between Parallel and Perpendicular Axis Theorem
Conclusion
The Perpendicular Axis Theorem is an important concept for solving moment of inertia problems in JEE Main, particularly for 2D objects like thin plates or lamina. It simplifies calculations by relating the moment of inertia about a perpendicular axis to those of two in-plane axes.
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FAQs on JEE Main 2025: Physics Perpendicular Axis Theorem Proof, Formula and Derivation
1. Why is the Perpendicular Axis Theorem not applicable for 3-D objects?
The perpendicular axis theorem is not applicable for 3D objects in the case of a planar object in the x-y plane.
However, the perpendicular axis theorem doesn't work for three-dimensional objects because the equation is derived from the assumption that the object is planar.
2. What are the applications of the Perpendicular Axis Theorem?
Some of the most important application of the perpendicular axis theorem is outlined here below:
With the help of the perpendicular axis theorem, the moment of inertia about a third axis can be calculated.
By using the perpendicular axis theorem, the moment of inertia for a three-dimensional object can be calculated.
3. The MOI of a thin Uniform Rod with Mass M and Length L about an axis perpendicular to the rod, through its centre, is I. What is the MOI of a rod on an axis perpendicular to the rod at its endpoint?
a. 4I
b. 2I
c. I/4
d. I/2
To solve this problem, we can use the Parallel Axis Theorem. This theorem states that if you know the moment of inertia (MOI) of an object about an axis passing through its center of mass, the moment of inertia about any parallel axis can be found using the formula:
$I = I_{\text{cm}} + Md^2$
Use the Parallel Axis Theorem: The distance between the center of mass and the new axis (at the endpoint of the rod) is $d = \dfrac{L}{2}$.
Applying the Parallel Axis Theorem:
$I_{\text{endpoint}} = I_{\text{center}} + M \left(\frac{L}{2}\right)^2$
$I_{\text{endpoint}} = I + M \times \frac{L^2}{4}$
We know the moment of inertia about the center is II: So, we get:
$I_{\text{endpoint}} = I + \frac{ML^2}{4}$Final expression: The moment of inertia at the endpoint is four times the moment of inertia at the center:
$I_{\text{endpoint}} = 4I$
4. When do we use a Parallel Axis Theorem?
It is used to calculate the moment of inertia of the area of a rigid object, whose axis is parallel to the axis of the object, and passes through the centre of gravity of the object.
5. How to manage time while writing a Physics Exam?
Many students say that though they went prepared to give their Physics exam, due to the lengthy and tricky questions, it became difficult. To avoid such an experience, students are recommended to practice writing faster and memorizing the keywords. Also, working out objective type questions beforehand saves time in analysing and answering them. Lastly, students are advised to divide their time into sections namely A, B, C and D and attempt all the questions without fail.
6. How is the Perpendicular Axis Theorem derived?
The Perpendicular Axis Theorem derivation is based on the application of the Pythagorean theorem and the integration of mass elements in the plane. The proof shows that the sum of the moments of inertia about the X and Y axes in the plane equals the moment of inertia about the Z-axis, which is perpendicular to the plane.
7. What is the formula for the Perpendicular Axis Theorem?
The Perpendicular Axis Theorem formula is:
$I_z = I_x + I_y$
Where $I_z$ is the moment of inertia about the axis perpendicular to the plane, and $I_x$ and $I_y$ are the moments of inertia about the X and Y axes, respectively, lying in the plane of the object.
8. What is Perpendicular Axis Theorem proof?
The Perpendicular Axis Theorem proof involves using the concept of mass distribution in a plane. By considering an infinitesimal mass element $dm$ at coordinates $(x, y)$, and using the definition of moment of inertia, the theorem is derived by summing the contributions of all mass elements in the plane, resulting in the relation:
$I_z = I_x + I_y$
9. Is there a diagram for the Perpendicular Axis Theorem?
Yes, the Perpendicular Axis Theorem diagram typically shows a 2D object in the XY-plane, with axes X, Y, and Z. The Z-axis is perpendicular to the plane, while the X and Y axes lie within the plane of the object. The diagram helps visualise the relationship between the moments of inertia about the different axes.
10. How does the Perpendicular Axis Theorem simplify moment of inertia calculations?
The Perpendicular Axis Theorem simplifies moment of inertia calculations for planar objects by allowing you to calculate the moment of inertia about an axis perpendicular to the plane, using only the moments of inertia about two perpendicular axes in the plane. This reduces the need for complex integration.
11. Can the Perpendicular Axis Theorem be used in all rotational motion problems?
The Perpendicular Axis Theorem is only applicable to two-dimensional objects lying in a plane and is most commonly used in planar rigid body rotation problems. For three-dimensional objects or non-planar rotations, different methods, such as the parallel axis theorem or the perpendicular axis theorem in 3D, would be more appropriate.
12. Define Perpendicular Axis Theorem.
The Perpendicular Axis Theorem states that for a plane (2D) object, the moment of inertia about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two perpendicular axes in the plane, intersecting at the same point.
