Mass Moment of Inertia for Composite Shapes

Hello! Welcome to the first lesson of our new module on the Planar Kinetics of Rigid Bodies.

In the previous module, we finished our study of kinematics by learning how to describe motion, including the special case of rolling without slipping. Now, we shift our focus to kinetics, where we analyze the forces and moments that cause this motion.

Our first step is to understand a crucial property of any rotating body: its mass moment of inertia. This property is the rotational equivalent of mass. Just as mass represents an object's resistance to linear acceleration ($F=ma$), mass moment of inertia represents its resistance to angular acceleration. Today, your goal is to learn how to calculate the mass moment of inertia for composite shapes using the parallel axis theorem. This skill is foundational for almost all rigid body kinetics problems, particularly in aerospace where complex components are often modeled as assemblies of simpler shapes.

1. What is Mass Moment of Inertia?

The mass moment of inertia, denoted by $I$, is a measure of an object's resistance to being spun up or slowed down. A larger $I$ means more torque is required to achieve the same angular acceleration. It depends not only on the object's mass but, crucially, on how that mass is distributed relative to the axis of rotation.

Formally, it's defined by the integral:
$I=\int r^{2}dm$
where $dm$ is an infinitesimal element of mass and $r$ is its perpendicular distance from the axis of rotation.

Given your electronics background, you can think of this as another fundamental property of a system, analogous to how resistance or inductance characterizes an electrical component's opposition to the flow of current.

Fortunately, just as with the area moment of inertia you studied in Module 3, we rarely solve this integral directly. Instead, we use pre-calculated formulas for standard shapes.

To get a solid grasp of the concept and its governing equation, please watch the beginning of the following video.

2. The Parallel Axis Theorem

The formulas for mass moment of inertia are typically given for an axis passing through the object's center of mass (or centroid). But what if we need to rotate the object about a different axis? This is where the Parallel Axis Theorem becomes essential.

The theorem states that the moment of inertia $I$ about any axis is the sum of the moment of inertia $I_{cm}$ about a parallel axis through the center of mass, plus a transfer term.

The formula is:
$I=I_{cm}+m{d}^2$
Where:

• $I_{cm}$ is the mass moment of inertia about the body's center of mass (the value you'd find in a table).
• $m$ is the total mass of the body.
• $d$ is the perpendicular distance between the axis you're interested in and the parallel axis through the center of mass.

For a clear, formula-based explanation and a simple application, let's look at a text resource.

3. Procedure for Composite Shapes

Most real-world objects, like an aircraft wing-flap assembly or a satellite with solar panels, are composite shapes. To find their total mass moment of inertia, we use the principle of superposition. The process is straightforward and leverages the Parallel Axis Theorem.

Step-by-Step Guide:

1. Decomposition: Break the complex object down into simple, standard shapes (rods, disks, plates, spheres, etc.). If there is a hole or cutout, treat it as a "negative" shape.
2. Locate Axes: Identify the overall axis of rotation for the entire assembly. For each individual shape, locate its center of mass.
3. Find $I_{cm}$: For each shape, find the formula for its mass moment of inertia about its own center of mass ($I_{cm}$). You will usually find these in a textbook table.
4. Measure Distance $d$: For each shape, measure the perpendicular distance ($d$) between its center of mass and the overall axis of rotation.
5. Apply Parallel Axis Theorem: For each shape, calculate its moment of inertia about the overall axis using $I=I_{cm}+m{d}^2$.
6. Summation: Add the moments of inertia of all the individual shapes to get the total moment of inertia for the composite body. If you have a cutout, subtract its moment of inertia.

$I_{total}=\sum I_i=\sum (I_{cm,i}+m_i{d}_i^2)$

4. Worked Example: Rod Assembly

Let's see this procedure in action with a classic example involving an assembly of two slender rods.

Common Formulas for $I_{cm}$:

As you saw in the video, having standard formulas is key. Here are a few of the most common ones for an object of mass $m$.

Shape	Axis	$I_{cm}$ Formula
Slender Rod (length $L$)	Perpendicular to rod, through center	$\frac{1}{12}m{L}^2$
Thin Circular Disk (radius $R$)	Perpendicular to disk, through center	$\frac{1}{2}m{R}^2$
Thin Rectangular Plate (sides $a,b$)	Perpendicular to plate, through center	$\frac{1}{12}m(a^2+b^2)$
Solid Sphere (radius $R$)	Through center	$\frac{2}{5}m{R}^2$

5. Composite Shapes with Holes

The principle of superposition also works for objects with cutouts. You calculate the moment of inertia as if the object were solid, and then you subtract the moment of inertia of the material that was removed.

The example in the video below demonstrates this perfectly.

Test your understanding!

Consider the pendulum shown below, which consists of a uniform slender rod and a uniform disk.

• The rod has a mass $m_{rod}=2$ kg and length $L=0.6$ m.
• The disk has a mass $m_{disk}=5$ kg and radius $R=0.2$ m.

Determine the mass moment of inertia of the entire assembly about the pivot point O.

Show answer

We will calculate the moment of inertia for each part about point O and then add them together.

1. Slender Rod:

• The axis of rotation (point O) is at the end of the rod. We can use the parallel axis theorem.
• The center of mass of the rod is at $L/2=0.3$ m from O. So, $d_{rod}=0.3$ m.
• The moment of inertia about its center of mass is $I_{cm,rod}=\frac{1}{12}{m}_{rod}{L}^2=\frac{1}{12}(2)(0.6{)}^2=0.06\text{ kg}\cdot {\text{m}}^2$.
• Using the parallel axis theorem: $I_{O,rod}=I_{cm,rod}+m_{rod}{d}_{rod}^2=0.06+(2)(0.3{)}^2=0.06+0.18=0.24\text{ kg}\cdot {\text{m}}^2$.
• (Alternatively, for a rod about its end, we can use the direct formula $I_{end}=\frac{1}{3}m{L}^2=\frac{1}{3}(2)(0.6{)}^2=0.24\text{ kg}\cdot {\text{m}}^2$).

2. Disk:

• The center of mass of the disk is at a distance $L=0.6$ m from the pivot O. So, $d_{disk}=0.6$ m.
• The moment of inertia of the disk about its own center of mass is $I_{cm,disk}=\frac{1}{2}{m}_{disk}{R}^2=\frac{1}{2}(5)(0.2{)}^2=0.1\text{ kg}\cdot {\text{m}}^2$.
• Using the parallel axis theorem: $I_{O,disk}=I_{cm,disk}+m_{disk}{d}_{disk}^2=0.1+(5)(0.6{)}^2=0.1+1.8=1.9\text{ kg}\cdot {\text{m}}^2$.

3. Total Moment of Inertia:

• Add the results for the rod and the disk.
• $I_O=I_{O,rod}+I_{O,disk}=0.24+1.9=2.14\text{ kg}\cdot {\text{m}}^2$.

Conclusion

You have now learned one of the most fundamental skills in rigid body kinetics. Calculating the mass moment of inertia is a prerequisite for analyzing how objects respond to torques, which is the focus of our upcoming lessons.

Key Takeaways:

• Mass moment of inertia ($I$) is a geometric property that quantifies an object's resistance to angular acceleration.
• For a composite body, the total moment of inertia is the sum of the moments of inertia of its parts, all calculated about the same axis.
• The Parallel Axis Theorem, $I=I_{cm}+m{d}^2$, is the critical tool used to shift the axis of rotation from an object's center of mass to the required axis for the assembly.

Next Lesson Preview:
In the next lesson, we will put this new skill to immediate use. We will introduce the three fundamental equations of motion for a rigid body in planar motion, which are the rotational equivalents of Newton's second law ($F=ma$). You will see that the mass moment of inertia, $I$, is a central component of the moment equation: $\Sigma M=I\alpha$.