Tennis racket theorem

The tennis racket theorem states that a freely rotating rigid body is unstable when it spins about its intermediate principal axis of inertia, producing repeated flips in orientation.

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Euler's equations of rigid-body motion: Differential equations describing the rotation of a rigid body in its body-fixed principal axes. Their torque-free solutions reveal why rotation about the intermediate axis is unstable.
Dzhanibekov effect: The repeated flipping of a freely rotating object about its intermediate principal axis, observed in spaceflight demonstrations. This name is commonly used for the theorem's striking observed manifestation.
Stable principal-axis rotation: Rotation about a body's largest or smallest principal moment of inertia, stable to small perturbations in torque-free motion. These two stable cases contrast with the theorem's unstable middle-axis rotation.
Spacecraft attitude dynamics: The study of how a spacecraft's orientation changes under rotation, torques, and control inputs. Spacecraft designers must avoid or manage unstable intermediate-axis rotation.
Principal axes of inertia: Three perpendicular directions in which a body's inertia tensor is diagonal. The theorem concerns rotation specifically about the axis with the middle principal moment.
Tennis racket: A handled frame with a tensioned mesh, whose shape gives it three distinct principal moments of inertia. A racket tossed in free flight can visibly flip when spun about its intermediate axis.
Forced rotation: Rotation changed by applied external torques. External torque can produce reorientation that resembles a flip but is not the theorem's instability.
Attitude control system: Hardware and algorithms that regulate a vehicle's orientation in space. Control systems can damp tumbling or prevent a spacecraft from entering unstable rotation.
Moment of inertia: A measure of how strongly a body's mass distribution resists angular acceleration about an axis. The ordering of moments of inertia determines which principal-axis rotations are stable.
Intermediate axis theorem experiment: A demonstration in which an asymmetric object is tossed while rotating about each of its principal axes. Comparing the tosses directly distinguishes stable spins from the unstable intermediate-axis spin.