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The perpendicular axis theorem and the parallel axis theorem are widely used in physics applications. These theorems help determine the value of the body’s moment of inertia lying in the perpendicular plane or the same plane. Further, there are multiple applications of the parallel and perpendicular axis theorem.
Let us understand all about these theorems, their prominent use cases, etc., followed by the main differences between the parallel axis theorem and perpendicular axis theorem. Hence, finding the proof of the perpendicular axis theorem becomes easy based on the definition, applications, and significant differences. Let us start with the quick definitions.
What is the perpendicular axis theorem?
This theorem gives details of the moment of inertia of a plane body around its axis. The perpendicular axis theorem states that any plane body’s moment of inertia about its axes is equal to the sum of the moment of inertia about any two perpendicular axes in the body’s plane which intersects the plane’s first axis. The only condition is that the plane should be perpendicular to the body’s plane.
What is the parallel axis theorem?
This theorem gives details of the body’s moment of inertia when lying in the same plane. It states that the moment of inertia of the body passing through its center is equal to the sum of the body’s moment of inertia about its axis passing through the center. It further states that the parallel axis theorem is equal to the product of the square of the distance between the two axes and the mass of the body.
Applications of parallel and perpendicular axis theorem:
Some of the main applications of the parallel and perpendicular axis theorem are:
Moment of inertia of a uniform rod about the transverse axis passing through its ends.
Moment of inertia of a thin uniform disc about its tangent perpendicular to its plane.
Moment of inertia of a thin uniform ring about its tangent perpendicular to its plane.
Moment of inertia of a thin uniform disc about its diameter.
Moment of inertia of a thin uniform ring about its diameter.
Difference between the parallel axis theorem and the perpendicular axis theorem
The key differences between the parallel axis theorem and perpendicular axis theorem are:
Parallel axis theorem | Perpendicular axis theorem |
It is applicable to any object. | It is applicable to the planar bodies or two-dimensional bodies only. |
Its formula is Io = Ic + m(d)2 Where, Io = moment of inertia of the object about point O. Ic = moment of inertia of the object about centroid C. m(d)2 = added moment of inertia due to the distance between O and C. | Its formula is Izz = Ixx + Iyy Where Izz = moment of inertia of the object about the 3D plane along the z-axis. Ixx = moment of inertia of the object about the 3D plane along the x-axis. Iyy = moment of inertia of the object about the 3D plane along the y-axis. |
It states that the moment of inertia of the object around any axis is equal to the sum of the moment of inertia of the object passing through the center of mass and the product of the object with the square of the perpendicular distance from the axis in consideration and the center of mass axis parallel to it. | It states that the moment of inertia of the object about any perpendicular axis is equal to the sum of the moment of inertia of the object about the other two mutually perpendicular axes lying in the object’s plane. |
Conclusion
Hence, it is easy to learn and understand the applications of the parallel and perpendicular axis theorem. Both are used widely in different calculations and therefore form an essential part of higher studies. Starting with the definition and the applications, it is easy to understand the importance of both theorems.
The main differences between the parallel axis theorem and the perpendicular axis theorem help go through the use cases of these theorems. Both deal with the relation between the moment of inertia of the body, the mass of the body, and the moment of inertia about the center. A quick look at the frequently asked questions helps students clear the main doubts.
