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Rotational Inertia of a Cube

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!
 

radou

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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.
 
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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482
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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.)?
 
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[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.
 
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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.
 
[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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Also, Ja4Coltrane, I don't understand what you mean. Could you please elaborate?
 

AlephZero

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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.
 
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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AlephZero

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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.
 

radou

Homework Helper
3,105
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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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