Deriving Moment of Inetia using just linear dynamics


by NANDHU001
Tags: deriving, dynamics, inetia, linear, moment
NANDHU001
NANDHU001 is offline
#1
Feb5-13, 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.
Phys.Org News Partner Physics news on Phys.org
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Mapping the road to quantum gravity
Chameleon crystals could enable active camouflage (w/ video)
nos
nos is offline
#2
Feb5-13, 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


Register to reply

Related Discussions
Deriving the second raw moment of the rice PDF Set Theory, Logic, Probability, Statistics 2
Deriving the first moment of area of semicircle Engineering, Comp Sci, & Technology Homework 3
Deriving moment of Inertia Introductory Physics Homework 2
Deriving the moment of inertia for a sphere Calculus & Beyond Homework 4
Deriving Moment of Inertia Introductory Physics Homework 2