AcademyRotational Dynamics
Academy
Conserving Angular Momentum
Level 1 - Physics topic page in Rotational Dynamics.
Principle
Angular momentum is conserved when net external torque about the chosen axis is zero.
Notation
\(L_i,L_f\)
initial and final angular momentum
\(\mathrm{kg\,m^{2}\,s^{-1}}\)
\(I_i,I_f\)
initial and final moment of inertia
\(\mathrm{kg\,m^{2}}\)
\(\omega_i,\omega_f\)
initial and final angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\tau_{\mathrm{ext}}\)
net external torque
\(\mathrm{N\,m}\)
\(K\)
kinetic energy
\(\mathrm{J}\)
Method
Derivation 1: Start from the torque law
Conservation is not a separate rule; it is the zero-external-torque case of the angular momentum law.
Torque law
\[\sum\tau_{\mathrm{ext}}=\frac{dL}{dt}\]
Zero external torque
\[\sum\tau_{\mathrm{ext}}=0\Rightarrow\frac{dL}{dt}=0\]
Conservation statement
\[L_i=L_f\]
Derivation 2: Use the fixed-axis form when valid
For a rigid body about one fixed axis, angular momentum is \(I\omega\), so changing mass distribution changes angular speed.
Initial angular momentum
\[L_i=I_i\omega_i\]
Final angular momentum
\[L_f=I_f\omega_f\]
Fixed-axis conservation
\[I_i\omega_i=I_f\omega_f\]
The graph shows the trade-off: if \(L=I\omega\) is fixed, reducing \(I\) increases \(\omega\).
Derivation 3: Do not assume energy conservation
Internal work can change kinetic energy even while angular momentum is conserved.
Rotational kinetic energy
\[K=\frac12I\omega^2\]
Energy warning
\[K_f-K_i\ne0\ \text{in general}\]
Rules
These are the compact results from the method above.
Conservation condition
\[\tau_{\mathrm{ext}}=0\Rightarrow L_i=L_f\]
Fixed axis form
\[I_i\omega_i=I_f\omega_f\]
Particle capture
\[mvr+I_i\omega_i=I_f\omega_f\]
Energy warning
\[K_f-K_i\ne0\ \text{in general}\]
Examples
Question
A skater changes from
\[I_i=4.0\,\mathrm{kg\,m^2}\]
to \[I_f=2.0\,\mathrm{kg\,m^2}\]
while spinning at \[3.0\,\mathrm{rad\,s^{-1}}\]
Find \(\omega_f\).Answer
Use
\[I_i\omega_i=I_f\omega_f\]
so \[\omega_f=\frac{4.0}{2.0}(3.0)=6.0\,\mathrm{rad\,s^{-1}}\]
Checks
- Conservation needs zero external torque about the chosen axis.
- Angular momentum may be conserved while kinetic energy changes.
- Internal forces can redistribute \(I\).
- Use the same axis before and after.