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Deriving Moment of Inetia using just linear dynamics

  1. Feb 5, 2013 #1
    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. jcsd
  3. Feb 5, 2013 #2


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    Yes, If you consider a mass being accelerated and rotates in a circle.
    Then the acceleration is:

    multiply both sides by r:

    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:
    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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