# Deriving Moment of Inetia using just linear dynamics

by NANDHU001
Tags: deriving, dynamics, inetia, linear, moment
 P: 22 Can moment of inertia be derived using just linear dynamics and calculus. Textbooks usually derive moment of inertia using energy equation and and analogy of 1/2mr^2w^2 with 1/2mv^2. I would like to know if it can be approached in a different manner using just linear dynamics.
 P: 30 Yes, If you consider a mass being accelerated and rotates in a circle. Then the acceleration is: $F=ma$ multiply both sides by r: $\tau=rma=r^{2}m\alpha$ where $\alpha$ is the angular acceleration. Take this sum of all masses: $\sum r^{2}dm$ Or another way: The force on a small element dm is: $dF=r\frac{d\omega}{dt}dm$ then the torque on this small mass dm is: $d\tau= rdF=r^{2}\frac{d\omega}{dt}dm$ integrating this over the total mass gives the total torque: $\tau=\int r^{2}dm\frac{d\omega}{dt}$ Hope it helps

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