Angular momentum is conserved when the net external torque on a system is zero. This is one of the most important
conservation laws in rotational dynamics.
\[
\vec\tau_{\mathrm{net}}=\frac{d\vec L}{dt}.
\]
If
\[
\vec\tau_{\mathrm{net}}=0,
\]
then
\[
\frac{d\vec L}{dt}=0,
\]
so angular momentum remains constant:
\[
\vec L_{\mathrm{initial}}=\vec L_{\mathrm{final}}.
\]
1. Angular momentum of a rotating rigid body
For a rigid body rotating about a principal axis,
\[
L=I\omega.
\]
Here, \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
2. Conservation equation for two rotating bodies
If two rotating bodies form an isolated system, the total initial angular momentum is
\[
L_i=I_{1i}\omega_{1i}+I_{2i}\omega_{2i}.
\]
The total final angular momentum is
\[
L_f=I_{1f}\omega_{1f}+I_{2f}\omega_{2f}.
\]
Conservation gives
\[
I_{1i}\omega_{1i}+I_{2i}\omega_{2i}
=
I_{1f}\omega_{1f}+I_{2f}\omega_{2f}.
\]
3. Bodies that stick together
In a completely inelastic rotational collision, two bodies stick together and rotate with a common final angular velocity:
\[
\omega_{1f}=\omega_{2f}=\omega_f.
\]
Then
\[
I_{1i}\omega_{1i}+I_{2i}\omega_{2i}
=
(I_{1f}+I_{2f})\omega_f.
\]
Solving for the common final angular velocity:
\[
\omega_f=
\frac{I_{1i}\omega_{1i}+I_{2i}\omega_{2i}}
{I_{1f}+I_{2f}}.
\]
4. Changing moment of inertia
If one object changes its moment of inertia without external torque, then
\[
I_i\omega_i=I_f\omega_f.
\]
Therefore,
\[
\omega_f=\frac{I_i\omega_i}{I_f}.
\]
This explains why a skater spins faster when pulling their arms inward. Pulling the arms inward decreases \(I\), so
\(\omega\) increases to keep \(L\) constant.
5. Explosions and separations
Conservation of angular momentum also applies to explosions and separations if the external torque is negligible.
For example, solving for \(\omega_{2f}\):
\[
I_{2f}\omega_{2f}
=
L_i-I_{1f}\omega_{1f}.
\]
\[
\omega_{2f}
=
\frac{L_i-I_{1f}\omega_{1f}}{I_{2f}}.
\]
A negative final angular velocity means the object rotates in the opposite direction to the chosen positive sense.
6. Angular momentum vs. rotational kinetic energy
In an isolated system, angular momentum is conserved. Rotational kinetic energy is
\[
K_{\mathrm{rot}}=\frac12I\omega^2.
\]
In a sticking collision, rotational kinetic energy is usually not conserved:
\[
K_i\ne K_f.
\]
The lost mechanical energy may become heat, sound, deformation, or internal energy.
7. Direction and sign convention
Angular momentum is a vector. In 2D rotation, the sign tells the direction:
- Positive \(L\): angular momentum points in the chosen positive direction.
- Negative \(L\): angular momentum points in the opposite direction.
In the usual \(x\)-\(y\) plane convention, counterclockwise rotation is positive and points in the \(+\hat{k}\)
direction, out of the page.
8. Worked example: two disks stick together
Disk 1 has
\[
I_1=0.8\ \mathrm{kg\,m^2},
\qquad
\omega_{1i}=20\ \mathrm{rad/s}.
\]
Disk 2 has
\[
I_2=0.4\ \mathrm{kg\,m^2},
\qquad
\omega_{2i}=0.
\]
They collide and stick together, so
\[
\omega_{1f}=\omega_{2f}=\omega_f.
\]
Initial angular momentum:
\[
L_i=(0.8)(20)+(0.4)(0).
\]
\[
L_i=16\ \mathrm{kg\,m^2/s}.
\]
Final angular momentum:
\[
L_f=(0.8+0.4)\omega_f.
\]
Conservation gives
\[
16=1.2\omega_f.
\]
\[
\omega_f=\frac{16}{1.2}.
\]
\[
\omega_f\approx13.3\ \mathrm{rad/s}.
\]
The final angular speed is lower than the first disk’s initial angular speed because the total moment of inertia
increases after the disks stick together.
Key idea: if there is no net external torque, total angular momentum stays constant even if moment of inertia and
angular velocity change.
