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Moment of Inertia of a Semicircle

Moment of Inertia of a Semicircle
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  • 0:00 Moment of Inertia
  • 0:38 Derivation
  • 4:29 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser
The moment of inertia of an object is the measure of its resistance to being rotated about an axis. In this lesson, we will derive the moment of inertia of a semicircle about an axis perpendicular to the semicircle through its radius.

Moment of Inertia

An object's moment of inertia is a measure of its resistance to being rotated about an axis. Think of a solid disk and a hoop rolling down a ramp. They both have the same mass and the same radius, and they're rotating about axes perpendicular to their centers. The object that has the least resistance to being turned will get to the bottom first. Which one will it be?

Don't worry, the answer will be revealed. But first, we must go through how to derive the moment of inertia of a half-circle about a radius perpendicular to its surface, as shown here:


Diagram 1. The axis comes out of the screen at point C
Diagram1


Derivation

The outline of our derivation is:

  1. Define a tiny strip of mass with differential width.
  2. Write an expression for area density for the whole semicircle and the tiny strips of differential widths.
  3. Add all of the individual strips using integral Calculus.

So, let's work our way through, starting with Step 1. We begin by defining a tiny strip of mass (dm) with differential width (dr) as shown here:


Diagram 2. The tiny strip is shown in black.
semicircle_dm


Now onto Step 2. We need an expression for area density, which is mass divided by area. We give area density the Greek letter sigma, σ. Density is an intensive property, meaning that it doesn't depend on the amount of the material. As long as the mass is uniform, its area density is the same whether you have the entire semicircle (macro-scale) or a small strip of differential width (micro-scale).

The macro scale area density is given in this equation:


Equation 1. The yellow, highlighted equation is the macro-scale area density.
Sigma_macro


The micro-scale area density is the same general equation, mass divided by area, but the mass is the tiny mass of the strip, dm, and the area is the rectangular shape of the strip shown here:


Diagram 3
strip2


This strip is the same strip in the semicircle, except we have taken it out of the circle and made it straight to show how we get the tiny area we need. The micro-scale area density is given in this equation:


Equation 2. The yellow, highlighted equation is the micro-scale area density.
sigma_micro


Notice that in both the macro and micro scale area density equations, σ is present.

Now onto Step 3. Step 3 is the longest step because it involves the summation of each strip's moment of inertia using integral calculus. The equation for the moment of inertia of an extended object is shown here:


Equation 3
I


There is a major issue with this equation. The variable r does not match the differential mass dm. We have to get dm in terms of dr, and this is where the area density equation comes in. Let's solve our second equation for dm.


dm


Now we can substitute σπrdr in for dm in our third equation, which will give us this new equation:


Equation 4. The yellow, highlighted equation is ready for integration because the only variable is r and it matches dr.
I_before_int


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