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Rotations of Rigid Object I
A Nexus mindmap
The summation shoud be an
integral for a continuous mass
0
0
0
Since it's motion in a circle
2
0
0
1
No!
3
0
1
2
Does the point on
the axis matter?
1
1
2
3
N.B. This argument holds for the angular
momentum about
any
point on the axis of
rotation, not just from the centre of mass.
1
0
1
2
0
4
0
0
1
It takes a little bit of thought to
see this pattern - it's not obvious!
3
So in 3D the moment of inertia
is no longer a scalar but in fact
a rank-2 tensor (a matrix)
4
What is meant here is the
integral of each component
of the matrix.
0
2
0
The matrix is symmetric!
1
3
0
0
Eigenvectors and eigenvalues!
1
Since it's symmetric
is can be diagonalised
1
Depends only on the geometry of the object
(well, the mass distribution across the object)
2
Have to be orthogonal
(for the cross product)
1
The moment of inertia will be for rotations of
the object around the origin (reference point).
0
3
Notes
2
1
This expression works
in 3D with
I
as a matrix
0
2
0
0
Note that we don't need to calculate
the first two columnt of
I
since we'll
be multiplying them by 0
0
1
0
0
1
0
2
0
1
3
1
0
1
0
2
3
0
0
0
1
2
3
1
0
2
(First 2 columns of
I
won't matter)
0
0
0
1
2
3
1
0
Orthonormal axes
1
0
1
2
ans
0
1
ans
0
0
ans
0
2
ans
0
3
What are the
pricipal axes?
4
symmetry
0
1
0
0
2
5
0
0
0
1
1
4
i.e. a flat object
0
3
2
e.g.
3
2
Rotations of Rigid Object I