Angular Momentum, Moment of Inertia
In prior posts we have discussed momentum, which we have assumed to be mass x velocity.
In a world where bodies can rotate, we need to recognise the difference between linear momentum and angular momentum.
When bodies are rotating, there is a rotational equivalent of momentum which is the product of angular velocity and something called the mass moment of inertia (MMOI).
Making sure we're talking about the right thing
Before we start exploring the concept of MMOI, it is helpful to list out a number of similar founding but very much different concepts.
First Moment of Area
Also commonly referred to as First Moment of Inertia
Measure of the distribution of an area relative to an axis. Can be used to find the centroid
Unit: m3.
Second Moment of Area
Also commonly referred to as:
Moment of Inertia of an Area
Area Moment of Inertia
Second Area Moment
Measure of the resistance to torsion
Unit: m4
Mass Moment of Inertia
Also commonly referred to as:
Moment of Inertia (without the Mass)
Angular Mass,
Second Moment of Mass
Rotational Inertia,
Measures resistance to rotational motion and angular (rotational) acceleration due to an applied torque (moment).
Unit: kg m2.
It is the third one that we are dealing with in this post.
What is the Moment of Inertia?
The moment of inertia I is also defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, that is:
Unlike mass, which depends only on amount of matter, moment of inertia is, in addition to the amount of matter, also dependent on the position of the axis of rotation and the shape of the matter. In otherwords, even when working exclusively in 2D, the moment of inertia for a certain shape would be different depending on the particular axis of rotation.
Calculating the moment of inertia
So how do we calculate the MMOI of our physics body? Wiki explains as follows.
The quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters.
In other words, restricting our discussions to 2D, moment of inertia I, for a point of mass m at distance r from the pivot point is simply:
The moment of inertia of a rigid body is the sum of the moments of inertia of all the points making up that rigid body.
More specifically, you can consider a body (in our 2D world, a lamina with a shape) as consisting of lots of small pieces that has mass dm. To find the mmoi of the entire lamina shape, we find equation for individual "elements" and integrate over the entire shape.
Let's work through some shapes to confirm our understanding.
MMOI of a hoop
We start with a thin circular loop of radius r and total mass M.
You can think of the loop as consisting of lots of points of mass dm. Every point would be at distance r, and hence the I of the individual point would be dm x squared(r). The mmoi of the loop would be the sum of all those individual points, hence intuitively you can guess that the mmoi of the loop is M x squared(r).
Let's be more rigorous about the way we calculate this. Again, think of the "loop" as consisting of lots of points, each with mass dm, distance r from the centre. Now, assume that each "point" has a "length" of dl, and the total "length" of the hoop is L. We can say that:
By definition,
which means that:
In order to find the mmoi for the loop, we need to integrate over the entire loop from limits of 0 to 2π, as follows.
MMOI of a "rod"
The rod is of length L and total mass M. In order to calculate the rod's mmoi around the axis which passes through its centre, assume that the rod is made up of elements of length dl, mass of dm.
As before, the relation between the total mass M and the total length L vs the elemental mass dm and elemental length dl is as follows:
It follows that:
Plug the above into the mmoi integral:
MMOI of a disc
In the first example, we talked about a point element with elemental mass of dm. In the second example we talked about an element of length dl and mass of dm.
In dealing with shapes with area, we introduce the concept of elemental area dA which has elemental mass of dm. The mmoi of the shape is the sum of the mmoi of the elemental area.
Thinking about the disc as concentric loops of "width" dx, distance r from the original.
If we denote the area of the circle by A, then we can say that:
We can also say that dA, which is the area of the infinitesimally thin hoop, is the product of the circumference and dx.
It follows that:
Now we can stick the above into the formula for MMOI.
This is quite hard work...
I will go onto deriving the mmoi for other shapes, but as this post is getting too long, I will finish this post off by introducing the below Wikipedia page which lists the formulae for various shapes.
The above covers a wide variety of the primitive shapes, including the "answers" to the shapes we have explored above. If you look closely, you will find that for some of the shapes, different formulae are given for different axis of rotation. It is critical to understand that unlike mass in linear motion, there is not one mmoi figure for a shape - the mmoi for a shape will be different depending on the axis of rotation.