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A rigid body consists of three thin uniform rods, each of mass $m$ and length $2a$, held mutually perpendicular at their midpoints. Choose a coordinate system with axes along the rods.

show that the moment of inertia is the same for any axis passing through the origin.

Show I calculated $I_{xx} = I_{yy} = I_{zz} = 2/3ma^2$ and I showed $I_{zx} = I_{yz} = I_{xy} = 0$, So $I = 2/3ma^2(\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\theta)$ where $\alpha,\beta,\theta$ are the angles between $x,y,z$-axis.

David K
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  • Notice that $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\theta)$ is a constant. You can work it out by taking the dot product of a unit vector on your arbitrary axis and the unit vectors of the $x,y,z$ axes. – David K Jun 04 '15 at 03:43
  • can you explain further I don't understand because I want to work them out and show rigorously why that is always constant? @DavidK –  Jun 04 '15 at 04:04
  • first of all how do we know how to represent our arbitrarily axis in order to take the dot product? @DavidK –  Jun 04 '15 at 04:25
  • Do you know what a unit vector is? How many unit vectors are parallel with any given axis through the origin? What are the coordinates of an arbitrary unit vector? What is the dot product of that unit vector with the $x$ axis? – David K Jun 04 '15 at 04:40
  • yes unit vector is a normalized vector with norm one along our i,j,k directions but I don't know how many unit vectors are parrallel with given axis through the orgin and is the representation of of arbitrarily unit vector $a\textbf{i} + b\textbf{j} + c\textbf{k}$? will the dot product of unit vector with x-axis evaluate to $atextbf{i}^2$ would it be possible if you could post complete answer explaining the logic that would be really nice. –  Jun 04 '15 at 04:52
  • $\sum cos^2 \alpha = 1$ ? If they can be treated as direction cosines. – Someone Jun 04 '15 at 05:25
  • http://mathworld.wolfram.com/DirectionCosine.html – David K Jun 04 '15 at 13:18
  • I understand now thank you ! –  Jun 04 '15 at 18:06

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I understand for the people interested it's by construction the directional vectors will have unit length of 1,so those sum will add up to 1 so that is why it doesn't change !