Parallel Perpendicular Axes Theorem

Introduction

The moment of inertia (MOI) of a body is determined by parallel axis theorem that is parallel to mass of object’s axial line and it also passes through the centre of gravitational force. The PAT suggests that the summation of moment of inertia of a body and product of the mass of a body and square of the axis distances defines the PAT. It helps in finding MOI of objects like rod, discs and a lot more. In this tutorial, the parallel axis theorem will be discussed further.

History of Parallel Axis theorem

The parallel axis theorem was postulated by Christiaan Huygens while he was studying the compound pendulum. The parallel axis theorem is also termed as Huygens-Steiner theorem because it was named after ‘Christian Huygens’ and ‘Jacob Steiner’

Parallel axis theorem

The parallel axis theorem of an object is used to assess the MOI of a solid and planar body that has an axis line parallel to the axis of the centre of gravity. According to Abdulghany (2017), the moment of inertia of the body determines its condition of existence on a particular plane.

The parallel axis theorem is determined by summation of the inertia of the body around the axis that is moving through midpoint of the mass along with product of the mass of a body multiplied by the square of the distance between two axes (library, 2022). As opined by Wang & Ricardo (2019), this aspect determines the core of the theorem.

This theorem applies to any solid body irrespective of its shape.

Figure 1: parallel Axes theorem

The formula for the Parallel axis theorem

The parallel axis theorem is associated with the solid bodies and is determined by adding up the moment of inertia of the body with the product of the mass of the body multiplied by the square distance between two axes (phy-astr, 2022).

I = I_x + Ma²

Here,

Here, ‘I’ = moment in the body, I_x = moment of inertia of the particular rigid body and M = mass of the solid object and a² determines the square of the distance between two particular axes.

Figure 2: Parallel Axis Theorem

Derivation of the Parallel Axis theorem

This theorem will be able to be derived in the following manner, let, I_x be related with the moment of inertia of a specific solid object and it is moving through the centre of the mass (let the axis be AB) and I will be moment of inertia of body that exists about the axis that is determined using A’B’ and it exists at a distance ‘a’ (phy-astr, 2022).

Now when considering the particle with a mass ‘m’ at a distance ‘r’ from the point that is the centre of the gravity, then it can be derived that A’B’ = r + a

Now

I = Σm ( r + a ) ²

I = ∑m (r² + a² + 2ra)

I = ∑m r² + ∑m a² + ∑2ra

I = I_x  +  a²  ∑m  +  2a∑m r

I = I_x  +  Ma² + 0

I = I_x  +  Ma²

Parallel Axis theorem of Rod

The parallel axis theorem of the rod will be determined mainly by moment of inertia of the rod. Moment of inertia of the rod is formulated by,

I = ⅓ ML²

Distance between the extreme end of the rod and the centre of the rod is calculated by,

a = L/2

Therefore, the parallel axis theorem is derived in the following manner

I_x = ⅓ ML² - M(L/2)²

I_x = ⅓ ML² - 1/4 ML²

I_x = 1/12 ML²

Moment of inertia

It has been previously studied that inertia is associated with the quality of the body exhibits in order to resist a specific state of the linear state of motion be it the object is in motion or is in a state of rest (Rempe et al. 2019). The moment of inertia of an object is indicated through the symbol Im and it is just a measure in which the body is existing in and its associated ability to resist the change in the linear state of motion or a rotational motion.

In both the cases of linear motion and rotational motion, the moment of inertia plays the exact role and thus both the conditions can be compared. Thus it can be established that the moment of inertia associated with a body is its capability to resist the alteration in its state of motion.

That Im = Mr², here r determines the particles perpendicular distance from its axis of rotation and M is related to mass of the body in a rotational motion. There is however certain factors associated with the moment of inertia, the moment of inertia is a scalar quantity and not a vector quantity, it is not constant for a body as it depends on the rotation of the axis, the moment of inertia of the object is dependent on the mass of the body (phy-astgr, 2022). Greater the mass of the body signifies greater moment of inertia. Subsequently, a greater moment of inertia results in a greater division of the mass from its axis.

Figure 3: Moment of inertia

Application of perpendicular and parallel axis theorem

Perpendicular along with parallel axis theorem is utilized in conjunction for finding moment of inertia of any rigid object about given axis. In order to calculate moment of inertia for any object in rotational motion, parallel along with perpendicular axis theorem are utilized. Parallel axis theorem is utilized when moment of inertia of any object about particular axis is known and is utilized for calculating moment of inertia about axis parallel to any other given axis. Perpendicular axis theorem is utilized for calculating moment of inertia of an object about third perpendicular axis in case of known moment of inertia of object for two perpendicular given axes.

Conclusion

The parallel axis theorem is applied in determining moment of inertia of the body about its parallel axis. This theorem was introduced by Christiaan Huygens when he was studying the principle of the compound pendulum. This theorem is commonly called Huygens - Steiner Theorem or simply Steiner theorem. Moment of inertia of specific body in an axial plane equals the summation of the moment of inertia of object and the product of the mass multiplied by the square of distance between the two axes involved.

FAQs

Q1. What is the moment of inertia?

Rotational inertia of any object is denoted as moment of inertia. The moment of inertia is a measure associated with the tendency of the object to resist a state of motion; it can be linear or rotational motion. Moment of inertia is relied upon mass of the object. Greater body mass indicates greater ability to retain the state of inertia.

Q2. When is the parallel axis theorem used?

This theorem is used when moment of inertia of particular body about one axis is obtained. However, it is required to find the moment of inertia is another axis that is parallel to first axis.

Q3. What is the general expression of the parallel axis theorem?

The parallel axis theorem is generally determined by the expression of I = I_x + Ma²

Here I_x represents moment of inertia of object that is rotating about axis that pass through centre of the mass. ‘M’ represents total mass of the object and a represents distance between axes through its centre of the mass of the particular object in consideration. I represent the moment of inertia of object that runs parallel to the rotational axis.

Q4. Who discovered the Parallel Axis theorem?

Parallel Axis theorem was discovered by Charles Huygens and Jacob Steiner. Therefore it is also called Huygens-Steiner Theorem or simply, it is called Steiner’s theorem.

References

Journals

Abdulghany, A. R. (2017). Generalization of parallel axis theorem for rotational inertia. American Journal of Physics, 85(10), 791-795. Retrieved from: https://aapt.scitation.org/doi/full/10.1119/1.4994835

Rempe, D., Sridhar, S., Wang, H., & Guibas, L. (2019). Learning generalizable final-state dynamics of 3D rigid objects. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (pp. 17-20). Retrieved from: http://openaccess.thecvf.com

Wang, J., & Ricardo, B. (2019). Squashing method for moment of inertia calculations. The Physics Teacher, 57(8), 551-554.Retrieved from: https://www.researchgate.net/profile/Bernard-Ricardo/publication/336707332_Squashing_Method_for_Moment_of_Inertia_Calculations/links/5dc3761a299bf1a47b1c03fa/Squashing-Method-for-Moment-of-Inertia-Calculations.pdf

Websites

library, 2022. About Inertia Intro: Parallel Axis Theorem. Retrieved from: https://pressbooks.library.upei.ca/statics/chapter/inertia-intro-parallel-axis-theorem/ [Retrieved on: June 7, 2022]

phy-astgr, 2022.About Moment of inertia. Retrieved from: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html [ Retrieved on : June 7, 2022]

phy-astr , 2022. About Parallel Axis Theorem . Retrieved from: http://hyperphysics.phy-astr.gsu.edu/hbase/parax.html [Retrieved on, June7, 2022]