Angular Momentum (3D)

What is Angular Momentum (3D)?

Angular momentum is a vector quantity — L = Iω (about a principal axis, where L and ω are parallel) — that points along the rotation axis according to the right-hand rule, not merely a scalar speed. A gyroscope, a spinning top, and a figure skater pulling her arms inward are all governed by the same principle: when net external torque is zero, L is conserved in both magnitude and direction. This simulation renders three scenarios in 3D: a precessing gyroscope, a skater changing moment of inertia by extending or retracting arms, and a perfectly inelastic collision between two rotating disks. The blue L vector and red torque vector τ update in real time so you can track how gravitational torque on the tilted gyroscope causes L to sweep out a cone — precession — rather than simply falling over. Use the camera drag to inspect the vector geometry from any angle.

Parameters explained

Angular Velocity(rad/s)

Angular Velocity sets the rotational speed ω of the spinning object in radians per second. In AP Physics 1 and HS-PS2 rotational dynamics, this slider is the direct link between motion and angular momentum because L = Iω (about a principal axis, where L and ω are parallel). With moment of inertia held fixed, increasing angular velocity increases the magnitude of L, making the spin axis harder to redirect. In a gyroscope or spinning top, a larger L also makes the same applied torque produce slower directional change, so precession becomes more gradual. Compare the Spinning Top and Gyroscope presets, then change only Angular Velocity to isolate how spin rate affects stability and precession.

Arm Extension(m)

Arm Extension changes the effective radius of the rotating mass distribution, which changes moment of inertia I. Mass farther from the rotation axis contributes more strongly because I depends on r², so increasing this slider makes the object harder to spin up or slow down. The Figure Skater preset is the clearest conservation test: with little or no external torque, pulling arms inward reduces I, so angular velocity must increase to keep L = Iω approximately constant (about a principal axis, where L and ω are parallel). This aligns with AP Physics 1 rotational inertia reasoning and HS-PS2 modeling because students connect a visible geometry change to a measurable change in motion.

Applied Torque(N·m)

Applied Torque sets the external twisting influence τ in newton-meters. Torque is not the same thing as angular momentum; it is the rate of change of angular momentum, τ = dL/dt. In this model, raising torque makes the L vector change direction more strongly, supporting the precession story for tops and gyroscopes. When torque is zero, angular momentum conservation is the focus: the Figure Skater preset shows how changing moment of inertia redistributes spin without needing an outside twist. When torque is nonzero, students can reason from HS-PS2-2 about how forces and interactions change motion over time.

Common misconceptions

• MisconceptionAngular momentum is just how fast something spins, so Angular Velocity and angular momentum mean the same thing.

  CorrectAngular Velocity is only one factor. Angular momentum is L = Iω (about a principal axis, where L and ω are parallel), so it depends on both rotational speed and moment of inertia. A slowly spinning object with mass spread far from the axis can have as much angular momentum as a faster compact object. Use Arm Extension to change I while watching why the product matters.

• MisconceptionWhen the Figure Skater preset speeds up, an outside force must be pushing the skater around.

  CorrectWith Applied Torque near zero, there is no outside twist causing the speed-up. Pulling arms inward reduces moment of inertia, so angular velocity increases to keep L = Iω conserved (about a principal axis, where L and ω are parallel). Internal muscle forces can change the skater's shape and rotational kinetic energy, but they do not create net external angular momentum.

• MisconceptionA larger Applied Torque always makes the object spin faster.

  CorrectTorque changes angular momentum, but the direction matters. A torque parallel to L can change spin speed; a torque perpendicular to L mainly redirects L and produces precession. In a gyroscope or top, torque can make the axis sweep around instead of simply increasing Angular Velocity.

• MisconceptionA gyroscope is stable because gravity stops acting on it.

  CorrectGravity still acts and can create torque about the support point. The key is that the spinning object already has angular momentum. The torque changes the direction of L over time, so the axis precesses rather than immediately tipping straight down.

• MisconceptionThe Spinning Top, Gyroscope, and Figure Skater presets are different laws of physics.

  CorrectThey are different examples of the same rotational dynamics ideas. The presets change Angular Velocity, Arm Extension, and Applied Torque to highlight different balances between L = Iω conservation (about a principal axis, where L and ω are parallel) and τ = dL/dt. The useful comparison is which parameter changed and how the angular momentum response followed.

How teachers use this lab

1. HS-PS2-2 precession investigation: start with the Gyroscope preset, record Angular Velocity, Arm Extension, and Applied Torque, then vary Applied Torque from 0 to 2 N·m while students describe how the interaction changes the direction of motion over time.
2. AP 5.D.1 conservation lab: use the Figure Skater preset with Applied Torque at 0 N·m. Have students move Arm Extension from 1.5 m toward 0.3 m and explain why Angular Velocity rises when moment of inertia decreases.
3. AP 5.E.1 torque discussion: compare the Spinning Top preset and Gyroscope preset, then hold Angular Velocity fixed while changing Applied Torque. Students use τ = dL/dt to connect a larger torque to faster angular momentum direction change.
4. One-variable control practice: assign teams one preset each, then allow them to adjust only Angular Velocity, only Arm Extension, or only Applied Torque. Each team writes a claim about which variable changed L's magnitude, which changed I, and which produced precession.
5. Misconception probe: before using the Figure Skater preset, ask students whether pulling arms inward should slow the skater because the body becomes smaller. They then test Arm Extension values and use L = Iω (about a principal axis, where L and ω are parallel) to revise the prediction.