Moment of Inertia

We all know that mass is loosely defined as the amount of matter in an object or mechanically, its measure of inertia. It is analogous to the mass of an object in linear motion but it is slightly different in one way. The moment of inertia of a point is the product of it mass and the square of its radius along with some constant that accounts for its shape. This can be seen as:

This gives the moment of inertia of a single point mass, not an entire object. Different objects have different moments of inertia based on their shapes.

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The moment of inertia of different shapes can be derived with a bit of calculus which we won't go into in-depth here because of how many different shapes have different moments of inertia.

The concept of moment of inertia because, while it is analogous to mass in linear settings, it is by no means the same thing. Mass is based on, well, mass. Moment of inertia is based on mass and radius from the pivot.

This distinction becomes significant in a scenario like the GIF below. All four rolling objects are assumed to have identical/equal mass m.However, the reason why the light green hoop finishes last is because it has a high moment of inertia, so its resistance to angular acceleration is lower(kind of like how linear mass measures how resistant an object is to linear acceleration), causing it to roll slower.

Its moment of inertia is highest here because all of its mass is concentrated far from the center. For the other objects, not all of the mass is far from the center so their moments of inertia are lower, meaning they roll faster down the ramp. If you're confused right now, head over to the page for rolling as that goes more in-depth on this idea.