I've just learned about moments of inertia in my physics I class, I=∫r2dm. The math involving moments of inertia in relation to torque and angular momentum is clear to me. I'm looking for some intuition towards where the definition/idea of moments of inertia comes from? Where do we obtain that the moment of inertia of a point mass I=mr2?
I'm interested in the logical development of these ideas.
Answer
Let's consider linear acceleration, governed by F=ma. If you have a system with many particles, and want to accelerate a system as a whole, the inertia depends on the total mass mtot=∑mi=∫dm
Now consider a single particle with mass m rotating at radius r. Using τ=Fr and a=rα, the relationship between torque and angular acceleration is F=ma→τr=mrα→τ=(mr2)α.
Now suppose you want to make a system rotate as a whole. Since each individual particle contributes a moment of inertia of mir2i, the total moment of inertia is just the sum, Itot=∑mir2i=∫r2dm.
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