Radius of Gyration: Solving Inertia Discrepancies

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SUMMARY

The discussion centers on the calculation of inertia for a non-uniform disk, specifically a spool. The correct formula for inertia is identified as I = m * k², where k represents the radius of gyration. The confusion arises from the standard formula I = 0.5 * m * r², typically used for uniform disks, which does not apply here due to the non-uniform density distribution of the spool. Participants emphasize the importance of using the effective radius provided in the problem statement to achieve accurate results.

PREREQUISITES
  • Understanding of rotational inertia and its formulas
  • Familiarity with the concept of radius of gyration
  • Basic knowledge of density distribution in materials
  • Ability to differentiate between uniform and non-uniform objects
NEXT STEPS
  • Study the derivation of the inertia formula for non-uniform objects
  • Learn about the application of radius of gyration in various engineering problems
  • Explore density distribution effects on rotational dynamics
  • Investigate other shapes and their inertia calculations, such as hollow cylinders
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Students in mechanical engineering, physics enthusiasts, and professionals dealing with rotational dynamics and material properties will benefit from this discussion.

theBEAST
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Homework Statement


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The Attempt at a Solution


Alright, so the solution manual shows that the intertia of the disk is:
I = m * k2, where k is the radius of gyration​

However why can't the inertia of the disk be:
I = 0.5 * m * r2, where r is the radius of the disk​
This is the formula for the inertia of a rotating disk... However when I plug in all my numbers I get a different inertia value when compared to the inertia value that I got using the radius of gyration.
 
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Because the spool is not a uniform disk. You don't need to know why it's not uniform, just accept that you are given the effective radius.
 
haruspex said:
Because the spool is not a uniform disk. You don't need to know why it's not uniform, just accept that you are given the effective radius.

Ah okay, I wish it said that the density is not uniformly distributed... But thanks!
 

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