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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.

          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.

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 ei that
rotate with the body, define the directions
u1 = (2√(2))/(3)e1 - (1)/(3)e3,
u2 = (√(2))/(3)e1 - (√(6))/(3)e2 - (1)/(3)e3,
u3 = -(√(2))/(3)e1 + (√(6))/(3)e2 - (1)/(3)e3,
u4 = e3
The locations of the masses are xi = aui for i = 1, …, 4.
Calculate the centre of mass and the components of the moment of inertia
tensor (with respect to the axes ei) at the centre of mass.
(ii) A solid ellipsoid has uniform density ρ.
e3
e2
e1
Points in the ellipsoid are described by
x = s(a sinθcosϕ e1 + b sinθsinϕ e2 + c cosθ e3)
for s ∈ [0, 1], θ∈ [0, π] and ϕ∈ [0, 2π] (ellipsoidal coordinates). The con-
stants a, b and c are the semi-major axes of the ellipsoid. The vectors e1, e2
and e3 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
f(x)dV = f(x(s, θ, ϕ))abcs^2 sinθ dsdθ dϕ
where the region of integration V is the image of the region D in ellipsoidal
coordinates.

Added by Christina M.

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