Moment of inertia about different axes

In summary, the conversation discusses the concept of moment of inertia and whether there is a body that has the same moment of inertia for all possible axes. An example of a solid sphere with the point located at its center is given as a possible solution. However, the conversation also mentions that the moment of inertia depends on the distribution of mass about the axis, and the reasoning behind the example of the hemisphere is questioned. Further discussion involves using integration and the perpendicular axis theorem to prove the moment of inertia for a square sheet.
  • #1
spaghetti3451
1,344
33
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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  • #2
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.
 
  • #3
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?]
 
  • #4
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)
 
  • #5
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.
 

1. What is moment of inertia about different axes?

Moment of inertia, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It depends on the object's mass and the distribution of its mass around a specific axis of rotation.

2. How is moment of inertia calculated for different axes?

The moment of inertia is calculated by integrating the squared distance of each particle in an object from the axis of rotation, multiplied by its mass. This calculation accounts for the object's mass distribution around the axis of rotation.

3. What is the difference between moment of inertia about different axes?

The moment of inertia can be different for different axes of rotation, depending on the object's shape and mass distribution. For example, a rod's moment of inertia will be different when rotated around its center compared to when rotated around one of its ends.

4. How does moment of inertia affect an object's rotational motion?

The moment of inertia affects an object's rotational motion by determining how much torque is needed to change its rotational velocity. Objects with a higher moment of inertia will require more torque to change their rotational motion compared to objects with a lower moment of inertia.

5. Can moment of inertia be changed for an object?

Yes, the moment of inertia can be changed for an object by either changing the object's shape or its mass distribution. For example, a figure skater can change their moment of inertia by extending or pulling in their arms while spinning, resulting in a change in their rotational speed.

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