Angular Momentum of a Rotating Body
Angular Momentum of a Rotating Body
Angular momentum is the rotational counterpart of linear momentum, and like linear momentum it is conserved in the absence of an external torque — which is why a spinning ice skater speeds up when pulling their arms in (reducing I increases ω to keep L constant), and why a spinning top or gyroscope resists changes to its orientation.Engineers use angular momentum to size braking systems for rotating machinery, to analyze gyroscopic effects in vehicles and aircraft, and to understand momentum-exchange devices like reaction wheels used to orient spacecraft without expending fuel.
Angular momentum is L = Iω, where the angular velocity ω is related to the rotation frequency by ω = 2πf. where I_rot is the moment of inertia of the rotating body, f_rot2 is its rotation frequency, omega_rot is the corresponding angular velocity, and L_ang is the resulting angular momentum.
Inputs
I_rot := 2 kg·m^2=2 kg·m^2
f_rot2 := 5 Hz=5 Hz
Convert the rotation frequency in cycles per second into angular velocity in radians per second — each full revolution sweeps through 2π radians.
omega_rot := 2·pi·f_rot2=31.42 Hz
Multiply the moment of inertia by the angular velocity to get angular momentum, exactly analogous to mass times velocity for linear momentum.
L_ang := I_rot·omega_rot=62.83 kg·m^2/s
Results
At 5 revolutions per second, the resulting angular momentum is substantial for this moment of inertia, reflecting both how fast the body is spinning and how much its mass resists changes in rotation. Because L is conserved without an external torque, stopping this rotation quickly (as in an emergency brake) requires a correspondingly large braking torque applied over a short time.
Assumptions & limitations
- Rigid body with constant moment of inertia during the spin.
- No external torque acting during the interval considered.
- Rotation about a fixed, well-defined axis.