M054 · Rotation / internal work

Angular-Momentum Skater

A skater analogue uses two symmetric sliding point masses on a rotating carrier. Prescribe a smooth inward or outward stroke, compare slow and fast pulls, then add a signed external torque. Inspect the actual radius, angular momentum, radial kinetic energy and independently integrated actuator work at any time.

Recorded time $0\,\mathrm{s}$

Actual slider radius $0\,\mathrm{m}$

Signed outward radial speed $0\,\mathrm{m/s}$

Instantaneous axial inertia $0\,\mathrm{kg\,m^2}$

Signed angular velocity $0\,\mathrm{rad/s}$

Angular acceleration $0\,\mathrm{rad/s^2}$

Axial angular momentum $0\,\mathrm{\frac{kg\,m^2}{s}}$

Rotational kinetic energy $0\,\mathrm{J}$

Radial kinetic energy $0\,\mathrm{J}$

Total modeled kinetic energy $0\,\mathrm{J}$

Integrated radial actuator work $0\,\mathrm{J}$

Integrated external torque work $0\,\mathrm{J}$

Actuator force on each mass · outward positive $0\,\mathrm{N}$

Combined radial actuator power $0\,\mathrm{W}$

Energy minus independently integrated work $0\,\mathrm{J}$

Angular-impulse balance defect $0\,\mathrm{\frac{kg\,m^2}{s}}$

PREDICT / PULL / ACCOUNT

Can momentum stay fixed while kinetic energy increases?

Signed axial and radial vectorsMomentum, angular velocity, torque and radial actuator forces have separate fixed scales.Release positionsGold wireframes mark the original mass centers; the physical motion has no displacement magnification.Mass of each slider

m

0.50\,\mathrm{kg}

Central carrier inertia

I_c

0.12\,\mathrm{kg\,m^2}

Release radius

r_0

0.65\,\mathrm{m}

Final radius

r_1

0.22\,\mathrm{m}

Stroke duration

T

3.00\,\mathrm{s}

Initial angular velocity

\omega_0

1.50\,\mathrm{rad/s}

External torque

\tau

0.00\,\mathrm{N\,m}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the gold release-radius handle and teal final-radius handle in the coordinate panel, or use the keyboard sliders. Editing creates a fresh trial. Each stroke starts after one second and ends with zero radial speed and acceleration; one further second is recorded. The violet dots in the work ledger are total actuator plus motor work. Faster motion adds transient radial kinetic energy. No masses are released, and required actuator forces have no hardware limit.

Ideal horizontal radial sliders with a prescribed quintic stroke, a fixed central carrier inertia and massless guides. This is a mechanical analogue, with no biomechanics, friction, actuator force limits or released masses. All mass remains attached. The rotating and radial kinetic energies both count. Angular momentum follows the external angular impulse; actuator work is obtained by integrating force times radial velocity, not by copying the energy change. The 3D marker size and carrier thickness are schematic.

Start with the essentials

Focus question: When external torque vanishes, what supplies the increase in kinetic energy?
One-sentence intuition: Axial angular momentum constrains the changing spin rate. The radial actuator supplies or removes kinetic energy; during the stroke, part of that energy is radial.

Core mathematical model

Axial inertia and external impulse

I(t)=I_c+2mr(t)^2,\qquad L(t)=I(0)\omega_0+\tau t,\qquad\omega(t)=\frac{L(t)}{I(t)}

Both point masses have the selected slider mass and remain attached. The central carrier inertia is fixed. Positive spin is counterclockwise from above.

Radial actuator force

F_r=m(\ddot r-r\omega^2),\qquad P_{\mathrm{act}}=2F_r\dot r

Force and radial speed are positive outward. This is the actual radial force on each mass, not a centrifugal force added to the inertial-frame equation.

Both kinetic-energy contributions

K=\frac12I\omega^2+m\dot r^2,\qquad\frac{\mathrm dK}{\mathrm dt}=\tau\omega+2F_r\dot r

The two sliders contribute radial kinetic energy in addition to their rotation. Work is integrated from the separately calculated physical powers.

Smooth radial programme

r=r_0+(r_1-r_0)(10s^3-15s^4+6s^5),\qquad s=\frac{t-1\,\mathrm{s}}{T}

The programme applies only during the stroke; radius is held constant before and after it. Speed and acceleration vanish at both joins. This is a prescribed motion, not a force-limited controller.

Common difficulties

Internal force can do work

Typical misconceptionNo external torque means kinetic energy cannot change.

Better mental modelRadial forces have zero axial torque but nonzero power when the masses move radially. Inspect the actuator-work curve.

A fast pull has another energy channel

Typical misconceptionAll kinetic energy is rotational throughout the stroke.

Better mental modelUse the fast-pull preset and inspect mid-stroke. Radial kinetic energy vanishes only when the radial motion stops.

An analogue has limits

Typical misconceptionThe display predicts a human skater’s muscle effort or released weights.

Better mental modelThere are no biomechanics, actuator limits, friction or ejection in this model. All masses stay attached, with an imposed smooth radius programme.

Run the experiment

01
Predict the final spin
With zero torque, compare release and final inertia, then complete the stroke.
What to observe: The inverse inertia ratio gives the spin ratio while axial angular momentum stays fixed.
02
Locate the work
Inspect the integrated actuator work and the total kinetic-energy change.
What to observe: Inward contraction increases final rotational energy; the actuator supplies it.
03
Compare fast and slow
Keep both endpoint radii fixed and change the stroke duration.
What to observe: At zero torque the endpoint spin and net work agree, but radial energy and peak force during the stroke differ.
04
Apply an external torque
Select the braking-torque preset and inspect the angular-impulse balance.
What to observe: Angular momentum now changes by external torque times elapsed time, including the one-second holds.