A gyroscope has I=0.01 kg·m², ω=20 rad/s, M=0.5 kg, d=0.1 m. What is the precession rate?

Ω = Mgd/(Iω) = (0.5)(9.8)(0.1)/((0.01)(20)) = 2.45 rad/s.

A skater with I=5 kg·m² at 2 rad/s pulls arms in to I=2 kg·m². What is the new ω?

L = Iω conserved: 5×2 = 2×ω_f → ω_f = 5 rad/s.

Disk A (I=0.5, ω=10) collides with stationary Disk B (I=0.5). What is ω_f for the combined system?

L_total = 0.5×10 + 0 = 5. I_total = 1.0. ω_f = 5/1.0 = 5 rad/s.

Pro 🔒~30 min

Angular Momentum (3D)

Conservation of angular momentum, precession, and gyroscopic effects

How it works

Angular momentum L = Iω is a vector quantity pointing along the axis of rotation (right-hand rule). In the absence of external torque, angular momentum is conserved: L_i = L_f. For a spinning ice skater, pulling arms in decreases I, so ω must increase to conserve L. A gyroscope tilted at angle θ from vertical experiences gravitational torque τ = Mgd·sinθ perpendicular to L, causing L to precess around the vertical axis at rate Ω = Mgd/(Iω). The faster the spin, the slower the precession. In collisions between rotating objects (e.g., two disks), the total angular momentum is conserved: I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω_f for a perfectly inelastic collision.

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Step-by-step

Select a scenario: gyroscope precession, ice skater spin-up, or rotational collision.
Adjust the spin rate and tilt angle.
In the gyroscope scenario, observe how the L vector sweeps out a cone (precession).
In the skater scenario, drag the arm extension slider to see ω change.
The 3D view shows L (blue arrow) and τ (red arrow) vectors in real time.
Rotate the camera with mouse drag.

Key formulas

$\vec{L} = I\vec{\omega}$ Angular momentum vector: moment of inertia times angular velocity vector
$\vec{\tau} = \frac{d\vec{L}}{dt}$ Net torque equals the time rate of change of angular momentum
$\Omega_{\text{prec}} = \frac{Mgd}{I\omega}$ Precession rate of a gyroscope: depends on weight, distance from pivot, spin angular momentum

Frequently asked questions

A gyroscope has I=0.01 kg·m², ω=20 rad/s, M=0.5 kg, d=0.1 m. What is the precession rate?: Ω = Mgd/(Iω) = (0.5)(9.8)(0.1)/((0.01)(20)) = 2.45 rad/s.
A skater with I=5 kg·m² at 2 rad/s pulls arms in to I=2 kg·m². What is the new ω?: L = Iω conserved: 5×2 = 2×ω_f → ω_f = 5 rad/s.
Disk A (I=0.5, ω=10) collides with stationary Disk B (I=0.5). What is ω_f for the combined system?: L_total = 0.5×10 + 0 = 5. I_total = 1.0. ω_f = 5/1.0 = 5 rad/s.