Perpendicular Axis Theorem

Perpendicular Axis Theorem, also known as the perpendicular axes rule, is a theorem in physics that relates the moments of inertia of a two-dimensional object about three perpendicular axes. It is one of those very fundamental physics principles that we use to understand how the object in question changes its rotational path perpendicular to the plane of its symmetry. The perpendicular axis theorem is useful in simplifying calculations of moments of inertia for objects with known moments of inertia about axes lying in their plane.

Through this article, we will simplify the Perpendicular Axis Theorem for better understanding, learn proof of it and compare it with the Parallel Axis.

What is Perpendicular Axis Theorem?

The perpendicular axis theorem in physics tells us that if we have a flat object and we know how it rotates around two axes lying in the same plane as the object, we can figure out how it rotates around an axis that’s sticking straight up out of the plane. Basically, if you add up how the object rotates around the x-axis and the y-axis, you’ll get how it rotates around the z-axis.

It helps us understand how hard it is to spin flat objects around. For instance, if we wish to know how hard it is to spin the coin around its edge, this theorem guides us in the same. It’s like a means of understanding how things rotate, which is essential for constructing vehicles and machines.

Perpendicular Axis Theorem Statement

Perpendicular Axis Theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane (z-axis) is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane of the object (x and y axes) and passing through the point where the perpendicular axis intersects the plane.

Perpendicular Axis Theorem Formula

The formula of Perpendicular Axis Theorem is given as

I_z = I_x + I_y

where,

• I_z is the moment of inertia of the object about the z-axis (perpendicular to the plane of the object).
• I_x is the moment of inertia of the object about the x-axis (lying in the plane of the object).
• I_y is the moment of inertia of the object about the y-axis (perpendicular to the x-axis and lying in the plane of the object).

Moment of Inertia

Moment of inertia is a measure of the difficulty of spinning something around. It depends on how the weight is balanced and how close it is to the center-line. Such as in case of merry-go-round when it is rotating, if all the kids sit near the middle, spinning becomes easier. But if they try to spin this to the edges, it gets more and more difficult. It is because the moment of inertia is higher when the mass is located far from the center. Consequently, the moment of inertia provides an answer to the question of how objects rotate and how they react to forces that try to either speed them up or slow them down. The moment of inertia formula is given as

I = ∑m_ir_i²

where,

• I is moment of inertia
• m is mass of particle
• r is distance from axis of rotation

Proof of Perpendicular Axis Theorem

The perpendicular axis theorem is based on this principle that the moment of inertia, when the axis which passes through the plane, is the sum of any two perpendicular axes which intersect each other.

Keep it in mind this plane lamina consisting of a large number of particles in the x-y plane is shown in the figure. Consider the particle with mass ‘m’, located at point P.

Let’s call PN and PN’ perpendicular to the x and y-axis, respectively.

The moment of inertia of the x-axis = my²

The total moment of inertia along the x-axis can be written as follows:

I_x = ∑my²—–(1)

The moment of inertia in the whole lamina about the axis is defined.

I_y = ∑mx²——-(2)

Similarly the moment of inertia of the whole lamina along the z-axes is found by,

I_z = ∑mr²

However, the r² = x² + y²

Therefore,

I_z = Σm(x² + y²)

From eq(1) and (2), we get:

I_z = ∑mx² + ∑my²

I_z = I_x + I_y

The single-axis theorem (perpendicular axis theorem) is used to calculate the moment of inertia of the body during which it is difficult to determine one principal axis of the body.

Difference between Perpendicular Axis Theorem and Parallel Axis Theorem:

Here’s a comparison between the Perpendicular Axis Theorem and the Parallel Axis Theorem:

Aspect	Perpendicular Axis Theorem	Parallel Axis Theorem
Definition	Represents the distribution of the objects mass around the axis selected.	It is calculation of moment of inertia by the axis, which is coincide to the object’s axis of rotation.
Orientation of Axis	Along the directions of the object or perpendicular to its plane.	the object image on the screen is parallel with the plane of that object.
Applicability	Able to be applied to for planar objects with uniform mass distribution.	Suitable for the trajectory calculation of any objects, of any shape and their mass distribution.
Use Cases	Typically, they are used in the designs of planar structures like the beams, the disks, and the plates.	Among the popular materials used in fabrication are two dimensional shapes and irregular forms.
Simplification of Calculations	One of the main objectives is to decrease algorithms to a perpendicular axis that calculates planar objects.	Generalize the moment of inertia computation around any axis by adding a coefficient that is distance from the center of mass over mass.

Application of Perpendicular Axis Theorem

The applications of perpendicular axis theorem is discussed below:

1. Calculating moment of inertia: The Perpendicular Axis Theorem is a primary method of determining the moment of inertia of objects with homogeneous mass symmetry. This comprises of objects, for example, such as rods, plates, or boss, whose axis of revolution goes through in the plane of the objects themselves.

2. Engineering design: In the Perpendicular Axis Theorem, engineers apply it to designing and creating structures and mechanics where rotational movement is involved. For instance, in the case of revolving part comprising flywheels or gear boxes, the moment of inertia plays a key a dynamic stability and efficiency.

3. Analysis of mechanical systems: In the daily mechanical engineering classrooms, the Perpendicular Axis Theorem is used to simulate the mechanical stiffness of the rotating parts. With an exact determination of moment of inertia, directors can predict how the system will react to the actions of applied force and torque.

4. Vehicle dynamics: Auto engineers use the Perpendicular Axis Theorem when they perform the rotational movement studies for car gear, wheels, axles, and transaxle. Understanding moment of inertia helps when aerodynamic design of vehicle is done with optimized handling, stability and energy efficiency.

5. Medical applications: Biomedical engineers likely employ rotational dynamics principles including the Perpendicular Axis Theorem in device engineering including prostheses, orthopedic implants and rehabilitation machinery. Using the concept of mass distribution and the forces of rotation during engineering design, they have created devices that not only assure mobility for the patient but also increase their health.

Perpendicular Axis Theorem: Conclusion

To conclude, the Perpendicular Axis Theorem is an important principle in physics, which is used to analyze rotational motion and to calculate moment of inertia. Its applications are very much extended, therefore making it a core physics content for schools. Through understanding the fundamentals of this theorem students can widen their knowledge of rotational dynamics and its practical examples.

Perpendicular Axis Theorem FAQs

What is the Perpendicular Axis Theorem?

The Perpendicular Axis Theorem is the physical principle that is used to calculate a planar object’s moment of inertia, in which the axis of the object is perpendicular to the object’s plane.

Where is Perpendicular Axis Theorem used?

Perpendicular Axis Theorem (PAT) is applied in the case of disk, plate, or beam and in analysis of their moment of inertia about the axis perpendicular to plane of the object.

How is Perpendicular Axis Theorem different from Parallel Axis Theorem?

The Perpendicular Axis Equation and the Parallel Axis Equations respectively are two formulas used to compute the moment of inertia about an axis perpendicular to the plane of the object and axis parallel to the plane of the object respectively.

What are the limitations of Perpendicular Axis Theorem?

As the Perpendicular Axis Theorem is valid for simple objects only with an uniform planar shape and mass distribution. It may not be able to handle objects with complex shapes, such as irregular or non-uniformly distributed masses, as additional methods may be needed for those cases.

Is the Perpendicular Axis Theorem useful in real-world applications?

The Perpendicular Axis Theorem is applicable in many including fields of engineering, physics, astronomy, and biomechanics. It is deployed in designing structures, in examining mechanical systems, predicting the operation of turbines, and tuning the sports equipment, as well as many other cases.