Hollow cube - moment of inertia

In summary, the conversation discussed how to calculate the moment of inertia of a hollow cube. The suggested method was to subtract the moment of inertia of the inner cube from the moment of inertia of the outer cube. The mass of the hollow cube was also mentioned and it was noted that it can be found using its volume and the given density. Finally, a formula for the moment of inertia of the hollow cube was proposed and confirmed by the participants of the conversation.
  • #1
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Homework Statement


How to calculate moment of inertia of hollow cube.

Homework Equations



The Attempt at a Solution


I guess that subtracting the moment of inertia of the inner cube from the moment of inertia of the outer cube is wrong.
Even it is close to solution, what mass to put in formula?
 
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  • #2
A good idea to subtract away the inner cube!
The mass of the hollow cube will appear in your answer; if you don't know it just leave it as m.
 
  • #3
I know mass of hollow cube, and it is m...
Moment of inertia of cube is m a^2/6.
So moment of inertia of hollow cube is m1 a^2/6 - m2 b^2/6...
I need help with m1 and m2...
 
  • #4
The density of the initial solid cube is a constant, so the mass is proportional to the volume. The volume of the removed cube and the volume of the remaining cubic shell are calculable in terms of a and b.
 
  • #5
I know that, but I don't know how to do it :(
Thanks for your answers!
 
  • #6
What is volume of the cube with an edge length a?
 
  • #7
? a^3
 
  • #8
Good. What is its mass (assuming density d)?
 
  • #9
d a^3
 
  • #10
So where is the problem?
 
  • #11
So...
I(of hollow cube) = m * [tex]\frac{a^3}{a^3-b^3}[/tex] * [tex]\frac{a^2}{6}[/tex] - m * [tex]\frac{b^3}{a^3-b^3}[/tex] * [tex]\frac{b^2}{6}[/tex]

that is

[tex]\frac{m(a^5-b^5)}{6(a^3-b^3}[/tex]

Is that ok?
 
  • #12
anyone?
 
  • #13
I don't get it. What is ma^3/(a^3-b^3)?
 
  • #14
Mass of hollow cube is given. So that is mass of full cube, before removing inner cube with edge length b.
 
  • #15
Ah, OK.

Looks OK to me, but I made so many stupid mistakes lately that I don't trust myself these days.
 
  • #16
Thank you...
 

What is a hollow cube?

A hollow cube is a three-dimensional shape with six square faces, all of equal size. It is similar to a regular cube, but with empty space inside.

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is dependent on the object's mass, shape, and distribution of mass.

How is moment of inertia calculated for a hollow cube?

The moment of inertia for a hollow cube is calculated by taking the sum of the moment of inertia for each individual square face. This can be calculated using the formula I = m[(a^2 + b^2)/12], where m is the mass of the face and a and b are the dimensions of the face.

Why is moment of inertia important for a hollow cube?

Moment of inertia is important for a hollow cube because it affects how the cube will behave when subjected to rotational motion. It can determine the cube's stability, how it will rotate, and how much force is needed to rotate it.

How can the moment of inertia of a hollow cube be changed?

The moment of inertia of a hollow cube can be changed by altering its mass, shape, or distribution of mass. For example, increasing the mass of the cube or changing the shape from a cube to a different polyhedron will change its moment of inertia.

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