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Deriving Moment of Inetia using just linear dynamics 
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#1
Feb513, 10:01 AM

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.



#2
Feb513, 11:37 AM

P: 32

Yes, If you consider a mass being accelerated and rotates in a circle.
Then the acceleration is: [itex]F=ma[/itex] multiply both sides by r: [itex]\tau=rma=r^{2}m\alpha[/itex] where [itex]\alpha[/itex] is the angular acceleration. Take this sum of all masses: [itex]\sum r^{2}dm[/itex] Or another way: The force on a small element dm is: [itex]dF=r\frac{d\omega}{dt}dm[/itex] then the torque on this small mass dm is: [itex]d\tau= rdF=r^{2}\frac{d\omega}{dt}dm[/itex] integrating this over the total mass gives the total torque: [itex]\tau=\int r^{2}dm\frac{d\omega}{dt}[/itex] Hope it helps 


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