Grade 12th passMechanics

Find the moment of inertia tensor for a solid cube of mass M and side a , rotating about a corner

Jasbir singh

8 Years agoGrade 12th pass

2 Answers

Saurav

8 Years ago

what i can figure out is we need to find moi about the diagonal(but it won’t affect the answer as we are dealing with uniform cube)

see both solutions

FIrst lets find moment Jz of a³ cube around its z-axis:

dJz = r² dm = (x² + y²) ρ dx dy dz
dJz = (x² + y²) aρ dx dy
Jz = integral (x² + y²) aρ dx dy
Jz = 2 integral x² aρ dx dy
Jz = 2a²ρ integral x² dx
Jz = 2/3 a²ρ [x³ from -a/2 to a/2]
Jz = 1/6 ma²

Since Jx=Jx=Jz, and x,y,z are obviously main axes of
tensor Jij of inertia, the elliopsoid of inertia of a cube
is perfectly degerate spherical.

Answer:
Moment of inertia of a cube around any axis
passing throug it center of mass is ma²/6

alternative

One of the big tricks that you find helpful is that the inertia matrix of a sphere is of the same form of that of a cube. I'll describe what this means.

For a sphere, imagine the moment of inertia (MoI) about three perpendicular axes through the centre of mass. Each MoI will be the same due to symmetry. Now, rotate the three axes you were just using, and the MoI about each axis is still the same

Now, the trick says you can do the same for a cube! So, you can easily calculate the MoI about an axis going through the centre of mass perpendicular to one of the faces of a cube. So you know the MoI about three perpendicular axes. Now, you can rotate the axes such that one aligns with the diagonal

And, like a cube, the MoI about the new axes will be the same as the MoI about the old axes, and so you can say: the moment of inertia about a body diagonal of a cube is the same as the moment of inertia about an axis through the centre perpendicular to one of the faces.

The reason this trick works is because a cube and a sphere have inertia matrices of the same form, and performing rotation operations on those matrices causes no change on either matrix, i.e. rotating axes does not change moment of inertia.

Best Of Luck...!!!!

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please approve

Forum Team

8 Years ago

Dear Saurav,

Please stop promoting your blog here. If you can answer queries of students here then it would be great otherwise guiding students to move to your blog is against forum policy. You will not be awarded for these points and your account may get blocked if you do not stop this. Please check the guidelines here – Forum point Policy – https://www.askiitians.com/forum-point-policy/

Thanks

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