Relation Between Angular Momentum, Moment of Inertia and Angular Velocity

Two particles of masses $m_1$ and $m_2$ are joined by a light rigid rod of length $r$. The system rotates at an angular speed $\text{ω}$ about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system, $L=\text{μ}{r}^2\text{ω}$, where, $\text{μ}$ is the reduced mass of the system defined as
$\text{μ}=\frac{m_1{m}_2}{m_1+m_2}$.

A uniform square plate of mass $2.0 \mathrm{kg}$ and edge $10 \mathrm{cm}$ rotates about one of its diagonals under the action of a constant torque of $0.10 \mathrm{N}-\mathrm{m}$. Calculate the kinetic energy(in $\mathrm{Joules}$) of the plate at the end of the fifth second after the start.

The density of a rod gradually decreases from one end to the other. It is pivoted at an end, so that it can move about a vertical axis through the pivot. A horizontal force $F$ is applied on the free end in a direction perpendicular to the rod. The quantities that do not depend on which end of the rod is pivoted are