(a) If a body is rotating about a fixed axis, then its angular momentum \(L\)
and its angular velocity \(\boldsymbol{o}\) are parallel vectors and
\(\mathrm{L}=\mathrm{le} \mathrm{s}\), where \(I\) is the (scalar) moment of
inertia of the body about the axis. However, in general, \(L\) and \(\omega\) are
not parallel and \(I\) in the equation must be a second-order tensor; let us
call it I. Find I in dyadic form in the following way: For simplicity, first
consider a point mass \(m\) at the point \(r\). The angular momentum of \(m\) about
the origin is by definition \(m r \times v\), where \(v\) is the linear
velocity.From Chapter \(6, v=\omega \times r .\) Write out the triple vector
product for \(L\) and from it write each component of \(\mathrm{L}\) in terms of
the three components of 6 . Write your results in matrix form
and in dyadic form
$$
\mathbf{L}=\mathbf{I} \cdot \mathbf{\omega}=\left(\mathrm{ii} l_{x
x}+\mathrm{ij} l_{x y}+\cdots\right)+\omega
$$
You should have
$$
I_{x z}=m\left(y^{2}+z^{2}\right), \quad I_{x y}=-m x y, \quad \text { etc. }
$$
For a set of masses \(m_{i}\) or an extended body, replace the cxpressions for
\(I_{z x+}, \operatorname{ctc}_{4}\), by the corresponding sums or integrals:
$$
\begin{aligned}
&I_{x x}=\sum m_{i}\left(y_{i}^{2}+z_{i}^{2}\right) \quad \text { or } \quad
\int\left(y^{2}+z^{2}\right) d m_{4} \\
&I_{x y}=-\sum m_{j} x_{i} y_{i} \quad \text { or }-\int x y d m_{4} \quad
\text { etc. }
\end{aligned}
$$
(b) Show that 1 is a second-order (Cartesian) tensor by expressing its
components relative to a rotated system \(\left[I_{s^{\prime}
x^{\prime}}=m\left(y^{2}+z^{\prime 2}\right)\right.\), etc. \(]\) in terms of \(x,
y, z\) using \((11.7)\) or \((11.11)\), and hence in terms of \(I_{x \pi}\), etc., to
show that I obeys the transformation equations \((11,13)\).
(c) Show that if \(\mathrm{n}\) is a unit vector, the expression
\(\mathbf{n}+\mathbf{1} \cdot \mathbf{n}\) gives the moment of inertia about an
axis through the origin parallel to n. Hint: Consider I rotated to a system in
which one of the axes is along \(n\).
(d) Observe that the I matrix is symmetric and recall that a symmetric matrix
may be diagonalized by a rotation of axes. The eigenvalues of the I matrix are
called the principal moments of inertia. Show by part (c) that they are
moments of inertia abour the new axes \(\left(x^{\prime}, y^{\prime},
z^{\prime}\right)\) relative to which \(\mathbf{I}\) is diagonal. These new axes
are called the principal axes of incrtia. For the mass distribution consisting
of point masses \(m\) at \((1,1,1)\) and \((1,1,-1)\), find the nine components of
1, and find the principal moments of inertia and the principal axes.
