3. A lamina of total mass $M$ is shown below. It has uniform mass density.
y
$(L, L/2)$
$(L, -L)$
x
A) (6 points) Calculate the inertia tensor for the coordinate system shown. Your final answer
should depend only on $M$ and $L$. Note: For this and all subsequent parts of this problem,
I encourage you to use Mathematica to evaluate integrals, but write down explicitly on
your written submission what integrals you are evaluating, including the limits of
integration. The same applies for any matrix and vector manipulations.
B) (6 points) Determine the principal moments and corresponding principal axes of the
lamina about the origin. No need to normalize your principal axes.
C) (2 points) The lamina is now set in rotation with constant angular speed $\omega$ about its lower
edge (the dashed line represents the rotational axis). From the vantage point of the figure,
the top point at $(L, L/2)$ initially rotates out of the page. Write down the angular velocity
vector $\vec{\omega}$ as expressed in the coordinate system shown.
D) (3 points) Calculate the angular momentum vector $\vec{L}$ associated with this rotation.
E) (3 points) Calculate the rotational kinetic energy $T$ associated with this rotation.
