Moment of inertia and rotational kinetic energy

AI Thread Summary
Moment of inertia quantifies an object's resistance to rotational acceleration, analogous to mass in linear motion. It is derived from the integral I = ∫ dm r², where dm is the mass element and r is the distance from the axis of rotation. This relationship connects torque (τ) and angular acceleration (α) through the equation τ = Iα, similar to how force (F) relates to linear acceleration (a) via F = ma. Additionally, rotational kinetic energy is expressed as KE = 1/2 Iω², paralleling the linear kinetic energy formula. Understanding these concepts is crucial for analyzing rotational dynamics in physics.
CrazyNeutrino
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I can't seem to understand moment of inertia. What does it mean and how is it derived ?
How does it relate to rotational kinetic energy.
 
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Hi CrazyNeutrino! :smile:

From the PF Library on moment of inertia …

Moment of Inertia … relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration.​

ie τ = Iα just like F = ma

(and KE = 1/2 Iω2 just like 1/2 mv2)

it is derived from I = ∫ dm r2
 
Why dm r^2?
 
CrazyNeutrino said:
Why dm r^2?

because for a point mass m at distance r from the axis, subjected to a force F,

the torque is τ = Fr, and the angular acceleration is α = a/r

τ = Iα means Fr = Ia/r

but F = ma (good ol' Newton's second law)

so mar = Ia/r

so I = mr2​

(and moment of inertia of the whole = sum of moments of inertia of the parts, giving us ∫ dm r2)
 

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