Angular Momentum — Spinning Disc
Change a spinning body's moment of inertia and watch its spin rate answer.
About this simulation
A skater pulling their arms in, or a second disc dropped onto a spinning one — either way L = Iω stays fixed while ω jumps as the moment of inertia changes. Bars for I, ω and L make the conservation visible, and a rotational-KE readout shows where energy goes when two discs couple.
How to use it
- Set the Initial spin ω₀; this starts the platform with a fixed angular momentum.
- Move the Arm radius slider to pull the point masses inward and watch the Angular velocity gauge climb while the Moment of inertia gauge falls.
- Check the Angular momentum value above the gauges: it stays the same as you change the arm radius.
- Switch Dead disc to 'Dropped on' and compare the Kinetic energy gauge before and after.
- Try the presets 'Arms pulled in', 'Arms flung out' and 'Heavy platform'.
What to notice
- With no external torque the angular momentum L = Iω stays constant, so a smaller moment of inertia means a faster spin.
- Moving the masses closer to the axis lowers I sharply because each mass contributes m r², which depends on the square of the distance.
- Rotational kinetic energy is not conserved: it rises when the arms are pulled in, because work is done pulling the masses inward.
- Dropping a non-spinning disc onto the platform raises I and lowers ω, and some kinetic energy is lost, just as in an inelastic collision.
- Mass that is let go while spinning carries its own angular momentum away, so the platform's ω does not change; only mass moving inward or outward, or non-spinning mass arriving, changes ω at fixed L.
The formulas
- L = Iω
- Angular momentum L is the moment of inertia I times the angular velocity ω.
- I = ½MR² + 2mr²
- Moment of inertia of a disc of mass M and radius R carrying two point masses m at distance r from the axis.
- KE = ½Iω²
- Rotational kinetic energy of a body with moment of inertia I spinning at angular velocity ω.
Questions
Why does a skater spin faster when she pulls her arms in?
While spinning on ice, a skater has almost no external torque on her, so her angular momentum L = Iω stays constant. Pulling her arms in brings mass closer to the axis and reduces her moment of inertia. To keep the product Iω the same, ω must increase. In the simulation, reduce the arm radius and watch the angular velocity gauge rise while the angular momentum value stays fixed.
What is the law of conservation of angular momentum?
If the net external torque on a system is zero, its total angular momentum stays constant. For a body rotating about a fixed axis this means I₁ω₁ = I₂ω₂, even when the body changes shape. It is the rotational counterpart of conservation of linear momentum, with torque taking the place of force. Divers, skaters and spinning planets all follow it.
Is kinetic energy conserved when a rotating body changes its moment of inertia?
Not in general. Since KE = L²/2I and L is fixed, reducing I increases the kinetic energy. The extra energy comes from the work the skater does pulling her arms inward against their tendency to fly outward. Increasing I does the opposite. In the simulation, the kinetic energy gauge rises as you pull the masses in, even though angular momentum stays the same.
What happens when a non-rotating disc is dropped onto a rotating disc?
Friction between the discs makes them spin together. These are internal torques, so the total angular momentum is conserved: I₁ω₁ = (I₁ + I₂)ω. The common angular velocity is smaller than before, and the final kinetic energy is less than the initial one, the rest being lost as heat. In the simulation, set Dead disc to 'Dropped on' and compare ω and kinetic energy.
