A spinning rotor resists rapid changes in the direction of its spin axis because it carries angular momentum.
This effect appears in gyroscopes, bicycle wheels, motorcycle wheels, spacecraft reaction wheels, flywheels, and
spinning projectiles.
\[
\vec L=I\vec\omega_s.
\]
1. Spin angular momentum
For a rotor spinning about its symmetry axis,
\[
L=I\omega_s.
\]
Here, \(I\) is the moment of inertia about the spin axis and \(\omega_s\) is the spin angular speed.
Increasing either \(I\) or \(\omega_s\) increases the angular momentum. A larger angular momentum vector is harder
to redirect.
2. Torque changes angular momentum
The fundamental rotational equation is
\[
\vec\tau=\frac{d\vec L}{dt}.
\]
A torque does not have to change the magnitude of \(\vec L\). In gyroscopic precession, the torque mainly changes
the direction of \(\vec L\).
3. Gravity torque for a tilted gyroscope
If the gyroscope has a mass \(M\), a lever arm \(r\), and a tilt angle \(\theta\) from the vertical, the perpendicular
lever arm is
\[
r_\perp=r\sin\theta.
\]
The gravitational torque magnitude is
\[
\tau=M g r_\perp.
\]
Therefore,
\[
\tau=M g r\sin\theta.
\]
4. Precession rate
In the high-spin approximation, the precession rate is
\[
\Omega_p=\frac{\tau}{L}.
\]
Since \(L=I\omega_s\),
\[
\boxed{
\Omega_p=\frac{\tau}{I\omega_s}
}.
\]
If the torque is gravitational,
\[
\boxed{
\Omega_p=\frac{M g r\sin\theta}{I\omega_s}
}.
\]
5. Stability dominance ratio
A useful way to judge gyroscopic stability is to compare the fast spin rate to the slower precession rate:
\[
D=\frac{\omega_s}{\Omega_p}.
\]
Substitute \(\Omega_p=\tau/(I\omega_s)\):
\[
D=\frac{\omega_s}{\tau/(I\omega_s)}.
\]
\[
\boxed{
D=\frac{I\omega_s^2}{\tau}
}.
\]
A larger \(D\) means stronger gyroscopic dominance. A small \(D\) means the torque can redirect the axis quickly.
6. Stability interpretation
The calculator uses this approximate scale:
| Dominance ratio |
Interpretation |
| \(D\ge 100\) |
Excellent gyroscopic dominance. Precession is very slow compared with spin. |
| \(30\le D<100\) |
Strong gyroscopic stability. |
| \(10\le D<30\) |
Moderate stability. Precession is visible but spin still dominates. |
| \(3\le D<10\) |
Weak stability. The axis can move noticeably under torque. |
| \(D<3\) |
Poor high-spin stability. The simple gyroscopic model may be unreliable. |
7. Disturbance deflection estimate
If a torque acts for a short time \(\Delta t\), the angular momentum direction changes approximately by
\[
\Delta\phi\approx\frac{\Delta L}{L}.
\]
Since
\[
\Delta L\approx\tau\Delta t,
\]
the approximate deflection angle is
\[
\boxed{
\Delta\phi\approx\frac{\tau\Delta t}{L}
}.
\]
This is a useful small-angle estimate for stability against short disturbances.
8. Solving design problems
The stability ratio
\[
D=\frac{I\omega_s^2}{\tau}
\]
can be rearranged to design a rotor.
Required spin speed for a target stability ratio:
\[
\boxed{
\omega_s=\sqrt{\frac{D_{\mathrm{target}}\tau}{I}}
}.
\]
Maximum allowable torque for a target stability ratio:
\[
\boxed{
\tau_{\max}=\frac{I\omega_s^2}{D_{\mathrm{target}}}
}.
\]
Required moment of inertia for a target stability ratio:
\[
\boxed{
I=\frac{D_{\mathrm{target}}\tau}{\omega_s^2}
}.
\]
9. Application: bicycle wheels
A spinning bicycle wheel has angular momentum. This contributes to resistance against rapid tilting and gives a
noticeable steering feel. However, bicycle stability is not caused by gyroscopic effects alone. Frame geometry,
trail, steering dynamics, rider control, and tire contact forces are also important.
10. Application: spacecraft reaction wheels
Spacecraft often use spinning wheels to store angular momentum. By changing the wheel speed, the spacecraft body
can rotate in the opposite direction to conserve angular momentum. External torques, such as magnetic or
gravitational-gradient torques, can slowly disturb the system.
11. Application: spinning projectiles and flywheels
A spinning projectile resists tumbling because its angular momentum vector is difficult to redirect. A flywheel
similarly stores rotational energy and angular momentum, which can help smooth rotational motion.
12. Worked example
Suppose a bicycle-style wheel has
\[
I=0.18\ \mathrm{kg\,m^2},
\qquad
\omega_s=62.8\ \mathrm{rad/s}.
\]
Let a tilted support create an approximate gravitational torque with
\[
M=2.0\ \mathrm{kg},
\qquad
r=0.20\ \mathrm{m},
\qquad
\theta=20^\circ.
\]
First compute the torque:
\[
\tau=M g r\sin\theta.
\]
\[
\tau=(2.0)(9.81)(0.20)\sin(20^\circ)\approx1.34\ \mathrm{N\,m}.
\]
Compute spin angular momentum:
\[
L=I\omega_s.
\]
\[
L=(0.18)(62.8)=11.3\ \mathrm{kg\,m^2/s}.
\]
Compute the precession rate:
\[
\Omega_p=\frac{\tau}{L}.
\]
\[
\Omega_p=\frac{1.34}{11.3}\approx0.118\ \mathrm{rad/s}.
\]
Compute the dominance ratio:
\[
D=\frac{\omega_s}{\Omega_p}.
\]
\[
D=\frac{62.8}{0.118}\approx532.
\]
Since \(D\) is large, the wheel has strong gyroscopic dominance under this approximate torque.
Key idea: gyroscopic stability is stronger when the rotor has large spin angular momentum and the external torque
is small.
