Exercise 1.
(i) A rigid body is made up of four equal masses (each of mass M/4) located at
the vertices of a regular tetrahedron. The four masses are held together by
a wire frame which has negligible mass. Using orthonormal vectors $e_i$ that
rotate with the body, define the directions
$u_1 = \frac{2\sqrt{2}}{3}e_1 - \frac{1}{3}e_3$,
$u_2 = \frac{\sqrt{2}}{3}e_1 - \frac{\sqrt{6}}{3}e_2 - \frac{1}{3}e_3$,
$u_3 = -\frac{\sqrt{2}}{3}e_1 + \frac{\sqrt{6}}{3}e_2 - \frac{1}{3}e_3$,
$u_4 = e_3$
The locations of the masses are $x_i = au_i$ for $i = 1, \dots, 4$.
Calculate the centre of mass and the components of the moment of inertia
tensor (with respect to the axes $e_i$) at the centre of mass.
(ii) A solid ellipsoid has uniform density $\rho$.
$e_3$
$e_2$
$e_1$
Points in the ellipsoid are described by
$x = s(a \sin \theta \cos \phi e_1 + b \sin \theta \sin \phi e_2 + c \cos \theta e_3)$
for $s \in [0, 1]$, $\theta \in [0, \pi]$ and $\phi \in [0, 2\pi]$ (ellipsoidal coordinates). The con-
stants $a$, $b$ and $c$ are the semi-major axes of the ellipsoid. The vectors $e_1$, $e_2$
and $e_3$ are orthonormal and rotate with the body.
Calculate the total mass, the centre of mass and the moment of inertia tensor
about centre of mass.
HINT: integrals in these ellipsoidal coordinates are
$\iiint_V f(x)dV = \iiint_D f(x(s, \theta, \phi))abcs^2 \sin \theta dsd\theta d\phi$
where the region of integration $V$ is the image of the region $D$ in ellipsoidal
coordinates.
