Is Calculating Moment of Inertia for a Disc with (x+dx)^2 Valid?

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SUMMARY

The discussion centers on the calculation of the moment of inertia (MI) for a disc, specifically addressing the validity of using the expression (x+dx)^2 in the calculation. Participants clarify that while one can multiply the mass of a thin ring by its radius squared to derive its moment of inertia, this approach must be adapted to account for the entire disc or non-infinitesimal rings. The conversation emphasizes the importance of considering the distribution of mass and the implications of using infinitesimal elements in the calculation.

PREREQUISITES
  • Understanding of moment of inertia concepts
  • Familiarity with calculus, particularly limits and infinitesimals
  • Knowledge of mass distribution in rigid bodies
  • Basic principles of rotational dynamics
NEXT STEPS
  • Study the derivation of the moment of inertia for various shapes, including discs and rings
  • Learn about the application of calculus in physics, focusing on integration techniques
  • Explore the concept of mass distribution and its effects on rotational motion
  • Investigate the differences between infinitesimal and finite mass elements in physics calculations
USEFUL FOR

This discussion is beneficial for physics students, mechanical engineers, and anyone involved in the study of rotational dynamics and moment of inertia calculations.

Arup Biswas
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In the picture the question marked spot is confusing me. Should the MI be calculated in this ugly way? That we do not consider the points in between and on dx! Why only x is taken as the distance? You may say take x+dx, resulting dx in the exapnsion and the dx from the mass to square and being negligible. But that too is not so simple. Can we write the whole mass as there is and just multiply it by (x+dx)^2. The definition does not say so. Please explain.
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It's not quite clear what you are asking.
Arup Biswas said:
Can we write the whole mass as there is
The whole mass of what?

Arup Biswas said:
and just multiply it by (x+dx)^2
Ah, I think you're saying: Can you take the mass of the thin ring and multiply it by its radius squared to get its moment of inertia. Sure. In fact, that's just what they are doing. But you want to do it in a way that you can calculate the moment of inertia of the entire disk or a non-infinitesimal ring.
 

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