Gyroscope

1. Premise and Key Insight

Premise:

• The angular velocity $\omega$ is sufficiently large.
• The spin angular momentum dominates: $L_{\text{wheel}} \gg L_z$.

Key Insight:
The torque $\vec{\tau}$ only affects the direction of the angular momentum $\vec{L}$, but it does not affect its magnitude.

2. Mathematical Derivation

The magnitude of the torque is given by:

$$\tau = Mgr$$

The magnitude of the angular momentum is:

$$L = I\omega$$

According to the rotational analog of Newton’s second law:

$$\vec{\tau} = \frac{d\vec{L}}{dt}$$

This implies that the change in angular momentum over a small time interval $dt$ is:

$$d\vec{L} = Mgr \, dt$$

Since the torque is perpendicular to the angular momentum, it causes the angular momentum vector to rotate by an infinitesimal angle $d\phi$ without changing its magnitude. From the vector triangle formed by $\vec{L}$ and $d\vec{L}$:

$$d\phi = \frac{dL}{L}$$

Substituting the expressions for $dL$ and $L$:

$$d\phi = \frac{Mgr}{I\omega} dt$$

3. Precession Rate

The precession rate (angular velocity of precession), denoted by $\Omega$, is the rate of change of the angle $d\phi$ with respect to time:

$$\Omega = \frac{Mgr}{I\omega} \quad \text{(precession rate)}$$

Important Conditions and Remarks

1. $\omega$ must be sufficiently large.
2. $\Omega$ is independent of mass $M$. This is because the moment of inertia $I$ is also proportional to the mass, so the mass terms cancel out in the numerator and denominator.