What is the rotational inertia of a cube when rotated about an edge?

In summary, the conversation discusses finding the rotational inertia of a cube of uniform density when rotated about an edge. The suggested method involves using the definition of rotational inertia, the parallel axis theorem, and converting to Cartesian coordinates for a triple integral. There is also a suggestion to consider the cube as a thin plate for further simplification.
  • #1
montreal1775
14
0
Does anyone know what the rotational inertia of a cube of uniform density is when it is rotated about an edge? Any help is appreciated!
 
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  • #2
montreal1775 said:
Does anyone know what the rotational inertia of a cube of uniform density is when it is rotated about an edge? Any help is appreciated!

I suggest you start with the definition of rotational inertia of a rigid body of uniform density.
 
  • #3
I=integral(r^2,m)

but then what do I do?
I'd use the parallel axis theorem but I don't know how to find the rotational inertia for a cube about it's center of mass.
 
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  • #4
montreal1775 said:
I'd use the parallel axis theorem but I don't know how to find the rotational inertia for a cube about it's center of mass.

You need to be able to derive the equation; you can't look it up (in your textbook, etc.)?
 
  • #5
[tex]
I = \int r^2 \,dm = \rho \int r^2 \,dV
[/tex]
where [itex]\rho[/itex] is the density of the cube (assumed to be uniform).

You can then convert r into a Cartesian equivalent, then split dV into dx, dy, dz and do a triple integral.
 
  • #6
I think you should consider the cube from above a plane with greater mass--that is-- 1/12(m)(2L^2). Then parallel axis it.
 
  • #7
Saketh said:
[tex]
I = \int r^2 \,dm = \rho \int r^2 \,dV
[/tex]
where [itex]\rho[/itex] is the density of the cube (assumed to be uniform).

You can then convert r into a Cartesian equivalent, then split dV into dx, dy, dz and do a triple integral.

So would it be [tex]
I = \iiint_{0}^{L} x^2+y^2+z^2 \,dxdydz
[/tex] where L is the length of a side?
 
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  • #8
Also, Ja4Coltrane, I don't understand what you mean. Could you please elaborate?
 
  • #9
r is the distance from the axis of rotation, not from the centre. If you are rotating about the z axis, then r^2 = x^2 + y^2.

Re Ja4Coltane's post, possibly he means the inertia of the cube about one edge is essentially the same as the inertia of a square about one corner. The thickness contributes to the mass, but it doesn't affect the geometric part of the formula.
 
  • #10
So is [tex] I = \rho \iiint_{0}^{L} x^2+y^2+z^2 \,dxdydz [/tex] the rotational inertial of a cube rotated about a corner?
 
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  • #11
What do you mean by "the rotational inertial about the origin"? I know what rotation inertia about a line is, and I know what the rotation inertia tensor about a point is (and it's got 6 independent components, not just one).

The rotation inertia about the edge defined by (x=0, y=0) is [tex] I = \iiint_{0}^{L} x^2+y^2 \,dxdydz [/tex] That's very similar to the inertia of a square about one corner.
 
  • #12
montreal1775 said:
Does anyone know what the rotational inertia of a cube of uniform density is when it is rotated about an edge? Any help is appreciated!

You can use the rotational inertia of a thin plate around the axis perpendicular to the plane of the plate to derive the moment of inertia of a cube around an edge.

Edit: this link may be of some use: http://hypertextbook.com/physics/mechanics/rotational-inertia/" [Broken].
 
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1. What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is a property that depends on an object's mass, shape, and distribution of mass.

2. How is rotational inertia calculated for a cube?

The rotational inertia of a cube can be calculated using the formula I = (1/6) * M * (a^2 + b^2), where I is the rotational inertia, M is the mass of the cube, and a and b are the length of two adjacent sides.

3. How does the shape of a cube affect its rotational inertia?

The shape of a cube plays a significant role in determining its rotational inertia. A cube with a larger mass and longer sides will have a greater rotational inertia compared to a cube with a smaller mass and shorter sides.

4. What is the relationship between rotational inertia and rotational motion?

Rotational inertia is directly proportional to rotational motion. This means that objects with a higher rotational inertia will require more torque to accelerate or decelerate their rotational motion compared to objects with a lower rotational inertia.

5. How can the rotational inertia of a cube be changed?

The rotational inertia of a cube can be changed by altering its mass, shape, or distribution of mass. For example, increasing the mass or length of the sides will increase the rotational inertia, while decreasing these factors will decrease the rotational inertia.

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