Moment of inertia about different axes

[spaghetti3451]

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?

 

[bsrishu]

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.

 

[Simon Bridge]

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?]

 

[bsrishu]

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)

 

[Simon Bridge]

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.