Illustrated Physics
physics, drawn
Concept

Hoop stress — where it appears

The tension in the wall of a spinning or pressurised ring, acting round its circumference. For a thin spinning hoop it is the material's density times the square of the rim speed, so a material's strength over its density sets the fastest it can turn.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The same liquid climbs a thin rod and is thrown off a thick one. The height of the free surface against distance from the axis, for rods of 10.0, 25.0, 60.0 millimetres radius turning at one revolution a second in the same polymer solution, with the Newtonian answer dashed. Two effects compete and they fall off at different rates: a hoop tension along the curved streamlines pulls the fluid inward and lifts it, falling as the fourth power of the distance, while the centrifugal term pushes it outward and lowers it, falling as the second. Near a thin rod the fourth power wins and the liquid climbs — 2.21 millimetres at the thinnest. Beyond a radius of 34.64 millimetres it does not, and the same fluid at the same speed is thrown outward exactly as water would be. The changeover is a property of the fluid and the rod together, so a demonstration that works on a glass stirring rod fails on a spoon handle.

The liquid that climbs the rod

Stir water with a rod and it is flung outward, leaving a dip at the centre. Stir a polymer solution and it climbs the rod instead. Nothing about the viscosity can produce that, however large it is made — what produces it is a second stress that appears in shear and has no Newtonian counterpart at all, growing as the square of the rate where the familiar one grows in proportion.

fluids · Rheology A gas pressed against the wall of a spinning tube. The density of uranium hexafluoride gas in a rotor spinning at a steady rate, relative to its density on the axis, on a logarithmic scale, against the distance from the axis as a fraction of the radius, for wall speeds of 300, 450 and 600 m/s at 300 K. In the turning frame the gas sits in the centrifugal potential −½mω²r² and settles into the same exponential that thins the atmosphere with height: n ∝ exp(mω²r²/2kT). At 600 m/s the density at the wall is e^25.4 — about 10¹¹ — times that on the axis; at 300 m/s, e^6.4. Almost all the gas lies in a thin layer against the wall, and the middle of the rotor is close to vacuum.

The spin that sorts by a difference of mass

In a spinning tube a gas settles against the wall the way the atmosphere settles towards the ground, with an exponential whose scale is set by the rotor's speed rather than by gravity. Heavier molecules settle more steeply, and the ratio of two isotopes changes from axis to wall by a factor that depends on the difference of their masses, not on the ratio. That one change from a ratio to a difference is why a spinning rotor separates uranium-235 from uranium-238 more than fifty times as strongly per stage as the porous barriers that first did it — and why the history of the machine is a history of how fast a material can turn before it flies apart.

mechanics · Circular motion

Named alongside it

The objects these essays reach for when they reach for this one.

Boltzmann distributionCentrifugal forceConstitutive lawCreeping flowEffusionFree surfaceIsotope separationMaterial functionNormal stressPolymer solutionRelaxation timeRod-climbing

All concepts