A particle has . It can have angular momentum about even while travelling along a straight line. For a whole body, separate motion of the centre from rotation about it:
The first contribution is orbital angular momentum relative to ; the second is spin about the centre of mass. For planar rotation, .
About a fixed inertial origin, . Integrating over a process gives angular impulse:
If only the axial component of external torque vanishes, only that component of angular momentum is assured constant. Gravity, for example, may exert no torque about a vertical axle while exerting torque about other axes. The same torque evolution law holds about the centre of mass for the angular momentum defined relative to it. For arbitrary moving origins, use the correction already discussed.
Before applying conservation, name the system: “wheel alone” and “wheel plus person” are different choices. The person exerts torque on the wheel, but that interaction is internal to their combined system. A fixed frictionless axle may exert large forces while contributing no torque about its own axis.
If the external torque is small but not zero, conservation is an approximation that needs a scale comparison. Estimate its angular impulse and compare it with the angular momentum of interest. A brief process can make a finite gravitational torque negligible, but a large impulsive support torque cannot be dismissed merely because the duration is short. Choosing an origin at an ideal pivot often removes that pivot's torque; it does not remove every other external torque.
Explore in 3D: Turntable and angular momentum
What happens when inertia changes?
At states with purely rotational motion, . Reducing inertia at constant angular momentum increases rotational kinetic energy. Internal chemical energy or an actuator supplies the increase; angular-momentum conservation does not imply mechanical-energy conservation. During inward motion, radial kinetic energy may also exist, so the endpoint formula assumes radial speeds vanish at the compared endpoints.
Watch the animation: Pulling arms in
Where does the extra kinetic energy come from?
A person and turntable together have constant inertia about a vertical frictionless axle. The person holds two point-like masses initially at radius , rotating at . The masses are pulled symmetrically to radius . Find the final angular speed and the increase in mechanical kinetic energy. All radial motion has stopped at each endpoint.
Hint: include both masses in each moment of inertia
Conserve axial angular momentum for person, turntable and masses together. Then calculate the two endpoint energies separately.
Full solution
Thus . With no change in gravitational potential energy and no losses, the person's internal energy supplies of mechanical work. The required metabolic expenditure would generally be greater; it is not determined here.
Check: Angular momentum remains in both states. If the final radius equalled the initial radius, neither the speed nor endpoint kinetic energy would change.
Choosing the point
Hinge: take about the hinge to remove the hinge force. Friction on a floor: take about any point on the floor, because friction acts along that line and has no arm. Free body: use the centre of mass or any fixed point, together with linear momentum.
Motor on a disc. When parts move relative to one another, write each part's angular momentum relative to the ground and include the orbital term of any body whose centre moves in a circle.