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:

$F=ma$
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