Moment of inertia about different axes

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Discussion Overview

The discussion revolves around the concept of moment of inertia, specifically exploring whether there are bodies that have the same moment of inertia about all possible axes or about all axes passing through a certain point. The scope includes theoretical reasoning and examples related to physics concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether any body can have the same moment of inertia about all axes and expresses uncertainty in providing an example or disproving the hypothesis.
  • Another participant suggests that a solid sphere has the same moment of inertia about all axes passing through its center, while also mentioning a hemisphere and a square sheet as examples for specific axes.
  • A different participant provides reasoning that moment of inertia depends on the distribution of mass about the axis, indicating a conceptual understanding of the topic.
  • There is a challenge regarding the hemisphere's moment of inertia, with one participant questioning the validity of the claim that it has the same moment of inertia about axes through its center and perpendicular to its plane surface.
  • Another participant mentions the need for integration to find the moment of inertia for the hemisphere and references the perpendicular axis theorem for the square sheet.
  • One participant expresses surprise at the claims made about the hemisphere and indicates a willingness to work through the calculations to verify the information.

Areas of Agreement / Disagreement

Participants generally express uncertainty regarding the existence of bodies with the same moment of inertia about all axes, with some proposing specific examples while others challenge those examples. The discussion remains unresolved with multiple competing views on the topic.

Contextual Notes

Participants note that the moment of inertia is influenced by mass distribution, and there are references to specific methods (integration, perpendicular axis theorem) that may be necessary to validate claims about certain shapes. The discussion does not reach a consensus on the examples provided.

spaghetti3451
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I found this problem in a book and am trying to answer it by myself.

1. Can you think of a body that has the same moment of inertia for all possible axes? If so, give an example, and if not, explain why this is not possible.

Solution: I can't think of any example or disprove the hypothesis. So I am wondering how I can answer this part.


2. Can you think of a body that has the same moment of inertia for all axes passing through a certain point? If so, give an example and indicate where the point is located.

Solution: An example is a solid sphere, with the point located at its centre. What do you think?
 
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1. I can't think of anybody having same moment inertia about any axis.
2. a). Sphere
b).Hemisphere also has same m.o.i about any axis passing through the centre of its plane surface.
C). Square sheet.
About any axis in its plane and through its centre.
 
1. reason why there's none: hint - moment of inertia depends on the distribution of mass about the axis.

2. sphere is useful - the important part is your reasoning.
[I would not have guessed the hemisphere one - any axis through the center of it's plane surface? Really? because I'd have thought an axis perpendicular to the surface would be different from an axis along it. Square sheet has an extra constraint besides "any axis through a point" though doesn't it?]
 
yes, m.o.i. Of hemisphere will be same along axis through the centre and the one perpendicular to it. If you want to find the m.o.i about the axis along the surface through its centre then you have to either see the symmetry or use the long integration method.
I have via done integration also and result comes out to be same.
For the square sheet, you can prove by perpendicular axis theoram.
(Not able to upload picture from mobile help)
 
Hmmm? I'm not doubting you - it's just not something I'd have guessed. Clearly I am now going to have to work it out since I also have not seen it in standard tables of moments of inertia. Well, it'll brush up my calculus and it doesn't look hard.

Time to hear from OP methinks.
 

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