AcademyRotational Dynamics
Academy
Gyroscopes and Precession
Level 1 - Physics topic page in Rotational Dynamics.
Principle
A gyroscope precesses because torque changes angular momentum's direction.
Notation
\(\vec L\)
spin angular momentum
\(\mathrm{kg\,m^{2}\,s^{-1}}\)
\(\vec\tau\)
external torque about the pivot
\(\mathrm{N\,m}\)
\(\Omega\)
precession angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\omega_s\)
spin angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(I_s\)
spin-axis moment of inertia
\(\mathrm{kg\,m^{2}}\)
\(d\)
pivot-to-center distance
\(\mathrm{m}\)
Method
Derivation 1: Find the torque about the pivot
The weight acts at the center of mass. If the center is displaced from the pivot, gravity produces a torque about the pivot.
Lever arm
\[d\]
Gravity torque
\[\tau=Mgd\]
Torque law
\[\vec\tau=\frac{d\vec L}{dt}\]
Derivation 2: Relate torque to slow precession
For fast spin and slow steady precession, torque mainly changes the direction of \(\vec L\), not its magnitude.
Spin angular momentum
\[L=I_s\omega_s\]
Direction-change rate
\[\tau=\Omega L\]
Slow precession
\[\Omega=\frac{\tau}{L}=\frac{Mgd}{I_s\omega_s}\]
Derivation 3: Check the approximation
The simple expression is a steady, slow-precession model. It should not be used when precession is comparable to spin.
Slow condition
\[\Omega\ll\omega_s\]
Faster spin effect
\[\Omega\propto\frac1{\omega_s}\]
Rules
These are the compact results from the method above.
Torque law
\[\vec\tau=\frac{d\vec L}{dt}\]
Spin angular momentum
\[L=I_s\omega_s\]
Gravity torque
\[\tau=Mgd\]
Slow precession
\[\Omega=\frac{\tau}{L}=\frac{Mgd}{I_s\omega_s}\]
Examples
Question
A rotor has
\[I_s=0.020\,\mathrm{kg\,m^2}\]
\[\omega_s=300\,\mathrm{rad\,s^{-1}}\]
\[M=1.5\,\mathrm{kg}\]
and \[d=0.10\,\mathrm{m}\]
Find \(\Omega\).Answer
Use
\[\Omega=\frac{Mgd}{I_s\omega_s}=\frac{1.5(9.8)(0.10)}{0.020(300)}=0.245\,\mathrm{rad\,s^{-1}}\]
Checks
- Precession direction comes from \(\vec\\tau=d\\vec L/dt\).
- Faster spin usually means slower precession.
- The simple formula assumes steady, slow precession.
- Torque changes the direction of \(\vec L\), not mainly its magnitude.