What is Moment of Inertia?
Moment of inertia measures how hard it is to change an object's rotation — the rotational equivalent of mass. It depends on both how much mass an object has and how far that mass is from the axis of rotation.
Moment of inertia is I = Σmr² for a system of point masses, or I = mr² for a single point mass a distance r from the axis, measured in kg·m².
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Step-by-step worked examples
A point mass of 3 kg sits 2 m from the axis. Find its moment of inertia.
I = mr² I = 3 × 2² I = 3 × 4 I = 12 kg·m²
A solid disk of mass 5 kg and radius 0.4 m rotates about its center (I = ½mr²). Find I.
I = 0.5 × m × r² I = 0.5 × 5 × 0.4² I = 0.5 × 5 × 0.16 I = 0.4 kg·m²
Two point masses on a light rod: m₁ = 2 kg at r₁ = 1 m, and m₂ = 3 kg at r₂ = 2 m from the same axis. Find the total moment of inertia.
I = Σmr² I = (2 × 1²) + (3 × 2²) I = 2 + 12 I = 14 kg·m²
Flashcards
Quick quiz
Q1.A point mass of 4 kg is 3 m from an axis. Find I.
Q2.Doubling the distance r from the axis multiplies I by:
Q3.Which shape has the largest I for the same mass and radius?
Q4.Moment of inertia is the rotational analogue of:
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Common mistakes
Moment of inertia only depends on mass. — Correct: It depends on both mass AND how far that mass is from the axis (r²).
I = mr² works for any rigid body. — Correct: I = mr² is only for a point mass; extended objects need I = ∫r² dm or shape-specific formulas.
Moment of inertia is the same about every axis. — Correct: I changes with the choice of rotation axis — the parallel axis theorem is needed to shift axes.
Doubling mass and doubling r have the same effect on I. — Correct: Doubling mass doubles I, but doubling r quadruples I because of the r² dependence.
FAQ
What is the formula for moment of inertia?
For a point mass, I = mr². For a system, I = Σmr²; for continuous bodies, I = ∫r² dm.
What is moment of inertia in simple terms?
It's how much an object resists changes to its rotation — heavier and mass farther from the axis both increase it.
How do you calculate moment of inertia for multiple masses?
Sum mr² for every mass in the system: I = m₁r₁² + m₂r₂² + …
Why does distance matter more than mass in moment of inertia?
Because I depends on r squared, so moving mass twice as far from the axis quadruples its contribution.