Physics - Classical Mechanics - Parallel Axis Theorem

Introduction

Hey it's a me again @drifter1! Today we continue with Physics and more specifically the branch "Classical Mechanics" to get into the Parallel Axis Theorem, that's quite useful for calculating Moments of Inertia (I). So, without further ado, let's get straight into it!

Parallel Axis Theorem

Let's consider a system of particles with a total mass M = Σm_i, that's rotating about a fixed axis O. The origin of our coordinate system will be placed at the center of mass (CM) of this system of particles. So, each general point of the system has some coordinate (x_i, y_i). In the same way the axis O also has some coordinate (a, b). All this can be described by a diagram like this one:

Of course from the definition of the center of mass (CM) we know that:

From the definition of the moment of inertia I_O = Σ_i m_i r_i²_O, we know that r is measured in respect to O, and not CM. But, the quantity r_i²_O can easily be written in respect to the center of mass as:

Substituting this into the equation of I we get:

From the two equation of the center of mass: x_CM and y_CM, we know that the x_i and y_i terms in this sum will go to zero. After that let's also split the sum into two parts: one with x's and y's and one with a's and b's. So, we end up with:

The sum (x_i² + y_i²) is simply equal to r_i², where r_i is now being measured from the CM. So, the first sum gives us the moment of inertia for the center of mass (I_CM). In the same way, we can also substitute the other quantity, cause the sum (a² + b²) is equal to R²_O, where R_O is the distance from O to CM. That way the second sum turns into the simpler term MR_O². So, in the end, we finally get the Parallel Axis Theorem or Huygens-Steiner Theorem equation:

To put it simply, this theorem tells us that the moment of inertia of any object about an axis is minimum when this axis is through its center of mass. For any axis parallel to that axis we have to add the term MR²_O or Md², where d is the distance of the axis of rotation (O) from the center of mass (CM). So, calculating the moment of inertia of common objects in any other axis of the object is easy, cause we just have to add the term Md² to the moment of inertia that has already been proven for the center of mass.

To understand the Theorem even better, let's now get into an example!

Example from physics.bu.edu

Let's consider a uniform rod of length L and mass M that's rotating about an axis through the center, perpendicular to the rod. How much is the moment of inertia for that axis?

In the previous article we said that the moment of inertia for the center of the rod, which tends to also be the center of mass, is given by:

Another common moment of inertia for such a rod is calculated at the end of the rod and given by:

Let's suppose that we don't know the moment of inertia for the center of mass (we can use it for validation), but we know the moment of inertia for the end of the rod! So, the Parallel Axis Theorem will be used for calculating I_CM:

The distance from the center to the end of the rod is simply L/2. By also substituting the moment of inertia that we know for the end of the rod we get:

We really end up with the equation that we already know for CM!!

RESOURCES:

References

http://www.public.asu.edu/~gbadams/sum00/parallelaxisT.pdf
http://hyperphysics.phy-astr.gsu.edu/hbase/parax.html
http://physics.bu.edu/~redner/211-sp06/class-rigid-body/parallelaxis.html

Images

Mathematical equations used in this article, where made using quicklatex.

Previous articles of the series

Rectlinear motion

Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
Projectile motion as a plane motion -> missile/bullet motion as a plane motion
Smooth Circular motion -> smooth circular motion theory
Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

Force and Newton's first law -> force, 1st law
Mass and Newton's second law -> mass, 2nd law
Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
Contact forces and friction -> contact force, friction
Dynamics of Circular motion -> circular motion dynamics, applications
Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
Contact force and friction examples -> exercises in force and friction
Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
Potential Energy Diagrams -> Internal Energy, Internal Work

Momentum and Impulse

Conservation of Momentum -> Momentum, Conservation of Momentum
Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
Collision Examples -> Various Elastic and Inelastic Collision Examples
Impulse -> Impulse with Example
Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis