Proving $\tau = I\alpha$ for Continous Mass Distribution

AI Thread Summary
To prove the relationship τ = Iα for continuous mass distribution, one must transition from the discrete case to an integral form. The moment of inertia I can be defined as an integral that accounts for the continuous distribution of mass rather than point-like particles. The proof relies on the concept that as the mesh value of the Riemann sum decreases, it converges to the Riemann integral, allowing for accurate approximation of torque and angular acceleration. This approach highlights the limitations of discrete models while demonstrating the validity of the continuous case. Ultimately, the proof reinforces the fundamental relationship between torque, moment of inertia, and angular acceleration in rotational dynamics.
pardesi
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how does one prove \tau=I\alpha for continious mass distribution where \tau is the net external torque along the axis of rotation I is the moment of inertia,and \alpha is the angular accelaration ...
i know the proof when the mass distribution is discrete...
 
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look at your discrete version and see how you can turn the sum into an integral ... eventually, i think the integral get absorbed in the definition of I (moment of inertia)
 
well taht doesn't happen ...because the discrete version necissates teh existence of point like particles...what does ahppen that thsi turns out to be a very good approximation...using the fact that as the mesh value of the riemann sum decrease it converges to the riemann integral; for a closed bounded function
 

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