The Gyroscope

First, we look at the change of angular momentum at the wheel with respect to time. This is equal to the total amount of torque in the system.

\begin{align*} \frac{d\vec{L_s}}{dt} &= \vec{\Omega} \times \vec{L_s} \\ \tau &= \Omega L_s \end{align*}

Next, we analyze the torque by looking at the Free Body Diagram. The weight of the wheel is the only force that contributes to the torque.

\begin{align*} \vec{\tau} &= \vec{l} \times M \vec{g} + \cancel{\vec{0} \times \vec{N}} \\ \tau &= M l g \end{align*}

Comparing these two equations gets us an equation for the angular velocity of the procession.

\begin{align*} \Omega L_s &= M l g \\ \Omega &= \frac{M l g}{L_s} \\ \Omega &= \frac{M l g}{I_0 \omega_s} \tag{8.2} \end{align*}