Deriving Moment of Inetia using just linear dynamics

NANDHU001

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.

nos

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