M054 · Rotation / internal work

Angular-Momentum Skater

Can angular momentum stay fixed while kinetic energy increases? Yes. With no outside torque the angular momentum stays fixed, so pulling mass inward makes the spin faster. Kinetic energy rises with it: halve the moment of inertia and both the spin rate and the kinetic energy double. The extra energy is the work done by whatever pulls the masses in, such as the arms of a skater.

Two point masses slide in and out on a rotating carrier, a mechanical stand-in for a spinning skater. Set a smooth inward or outward stroke, compare slow and fast pulls, then add an external torque in either direction. At any moment you can read the radius, the angular momentum, the radial kinetic energy and the actuator work, which is integrated separately.

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 in, then follow the energy

Can angular momentum stay fixed while kinetic energy increases?

Signed axial and radial vectorsAngular momentum, angular velocity, torque and radial actuator force arrows each have their own fixed scale.Release positionsGold wireframes mark where the mass centers started. The motion is drawn at true scale, with no magnified displacement.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\,\%

In the coordinate panel, drag the gold release-radius handle and the teal final-radius handle, or use the sliders from the keyboard. Any edit starts a fresh trial. Each stroke begins after one second and ends with zero radial speed and zero radial acceleration; the record then runs for one more second. In the energy and work plot, the violet dots show the total work of the actuators plus the motor. A faster stroke adds radial kinetic energy that lasts only while the masses move. The masses are never let go, and the actuator forces have no hardware limit.

The sliders are ideal and move horizontally along massless guides. Their radius follows a prescribed quintic stroke, and the central carrier's moment of inertia is fixed. This is a mechanical analogue: it has no biomechanics, friction, actuator force limits or released masses, and all the mass stays attached. Both rotational and radial kinetic energy are counted. Angular momentum changes by the external angular impulse. Actuator work is integrated from force times radial velocity, separately from the energy change. The 3D marker size and carrier thickness are schematic.

Physics tutorial

Conserved momentum does not mean conserved kinetic energy

BackgroundA skater spins faster by pulling mass toward the spin axis. Here two matching sliders move along radial guides on a rotating carrier. The force that pulls them points along the radius, so it exerts no torque about the axis, but it can still do work.

Why it mattersOpenStax University Physics section 11.3 connects a changing moment of inertia to angular-momentum conservation and internal work. This apparatus tracks the radial motion explicitly and adds an external torque as a separate test.

Start with the essentials

Focus question: With no external torque, where does the extra kinetic energy come from?
One-sentence intuition: Angular momentum about the axis sets how the spin rate changes as the masses move. The radial actuator puts kinetic energy in or takes it out, and during the stroke part of that energy is in the radial motion.

Core mathematical model

Moment of inertia and external angular 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 chosen slider mass and stay attached. The central carrier's moment of inertia is fixed. Spin is positive when it is counterclockwise seen from above.

Force the actuator applies along the radius

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 real radial force on each mass, written in the inertial frame with no centrifugal force added.

Rotational and radial kinetic energy

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

The two sliders carry radial kinetic energy on top of their rotation. The work comes from integrating each physical power, calculated separately.

Smooth radius schedule

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

The schedule applies only during the stroke; the radius is held constant before and after it. Radial speed and acceleration are zero at both joins. The radius follows this prescribed path; there is no controller with a force limit.

Common difficulties

Internal forces can do work

Typical misconceptionNo external torque means kinetic energy cannot change.

Better mental modelThe radial forces exert no torque about the axis, but they deliver power whenever the masses move in or out. Look at the actuator-work curve.

A fast pull puts energy into radial motion

Typical misconceptionAll of the kinetic energy is rotational throughout the stroke.

Better mental modelTry a fast pull and look at mid-stroke. Radial kinetic energy disappears only when the radial motion stops.

This is a mechanical model, not a skater

Typical misconceptionThe display predicts a real skater’s muscle effort, or what happens when weights are let go.

Better mental modelThe model has no biomechanics, actuator limits, friction or masses flying off. Both masses stay attached and follow a prescribed smooth radius schedule.

Run the experiment

01
Predict the final spin
With zero torque, compare the moment of inertia at release and at the end, then complete the stroke.
What to observe: The spin rate changes by the inverse of the inertia ratio, and the angular momentum about the axis stays fixed.
02
Find where the energy comes from
Look at the integrated actuator work and the total change in kinetic energy.
What to observe: Pulling inward raises the final rotational energy, and the actuator supplies that energy.
03
Compare a fast and a slow pull
Keep both end radii fixed and change the stroke duration.
What to observe: With zero torque, the final spin and the net work are the same, but the radial energy and the peak force during the stroke differ.
04
Apply an external torque
Select External braking torque and check the angular impulse.
What to observe: Angular momentum now changes by the external torque times the elapsed time, including the one-second holds.