Illustrated Physics
physics, drawn
Concept

Friction — where it appears

The force between two surfaces in contact along their common plane, which resists relative sliding and takes whatever value equilibrium demands until it cannot. Its independence of apparent area comes from the real contact area growing in proportion to the load, on asperities that flatten until they can carry it.

Named by 21 essays across 3 fields — each of them below, with the objects they name alongside it.

A block on a 27° incline. Free-body diagram of a block resting on an inclined plane: weight straight down, resolved into a component pressing into the surface and one pulling along it, with friction opposing the slide.

The slope, and the two directions that make it easy

An inclined plane looks like a harder problem than a flat one. Split the weight into two components chosen to suit the slope and it becomes an easier one.

mechanics · Free-body What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis.

The force that takes what it needs

Static friction has no value of its own. It supplies exactly what equilibrium demands and not a newton more, right up to the moment it cannot — which is the only instant in the whole business at which a coefficient of friction means anything at all.

mechanics · Friction The same steady pull, twice. Spring force divided by the normal load, against time, with the far end drawn away at 100 µm/s in both traces and every property of the contact the same. The soft holder at 0.4 k_c produces events every 3.75 s, each of which reaches 0.94 m/s — 9.4e+3 times the speed it is being pulled at. The stiff one at 1.6 k_c settles to a straight line and stays on it.

The chatter a stiffer holder removes

A brake squeals, a bow sounds a string, a fault slips in jerks. The usual explanation is that static friction exceeds kinetic friction — and that explanation, taken seriously, predicts the jerking would happen no matter how the thing were held. It does not, and what decides is a length nobody mentions.

mechanics · Friction The stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

The silo that does not weigh what it holds

Pour water into a tall vessel and the pressure at the bottom is the depth times the density times g, whatever the shape above it. Pour grain in and the floor stops learning anything new after the first couple of metres, because the walls have quietly taken the rest — and the length over which they take it contains no property of the grain at all.

fluids · Granular matter Rough contact grows as W^1.000, and one smooth bump as W^0.667. Real contact area against load, both logarithmic, three ways. Plastic flow of the asperities gives A = W/H, a slope of exactly one, which is the account Bowden and Tabor's argument uses. A single elastic sphere gives Hertz's answer, A ∝ W^(2/3), and a friction coefficient falling as W^(−1/3) — Amontons' law would be false. A rough elastic surface, with many asperities spread exponentially in height and integrated here rather than quoted, gives a slope of 1.0000, because pressing harder mostly recruits new contacts rather than enlarging existing ones. The dotted curve repeats that integration with a Gaussian spread of heights and gives 0.9695 — the upper tail of a Gaussian is nearly exponential, so the answer is nearly and not exactly linear. Amontons' law does not need plasticity. It needs roughness, it is exact for one distribution and good to three per cent for another, and it survives whichever way the individual junctions deform.

The grip that is not a coefficient

A friction coefficient is independent of load and of area, and the reason is usually given as plastic flow of the asperities. It is not — a rough elastic surface gives the same law, integrated here to a slope of 1.0000, because pressing harder recruits new contacts rather than enlarging old ones. What breaks the law is a contact that is smooth, or a material that dissipates in its bulk — where μ passes one and stops being a coefficient at all.

mechanics · Friction The slope a heap of grains settles at. The steepest slope a cohesionless heap can hold, against the friction between its grains. A slab of thickness h on a slope is pushed down it by the weight's along-slope component and held by friction acting on the weight's across-slope component, and both are proportional to the same ρgh — so the thickness cancels and the criterion is tan θ = μ. Checked here at thicknesses of 2 mm, 50 mm, 2000 mm, the ratio of driving to holding stress differs by 2.2e-16, which is zero. That is why the quantity is an angle: nothing about the size of the pile, the size of the grains, the density or the strength of gravity survives into it, and a heap of sand on the Moon stands at the same slope as one on Earth. The marked materials are glass beads at 24°, dry sand at 33°, crushed gravel at 40°. The band between the two curves is the hysteresis: a slope steeper than 31.0° will keep flowing once started, and one shallower than 35.0° will not start — so a pile has a range of stable angles rather than one, and which it is found at depends on how it was built.

The angle that does not know the size of the heap

Pour sand and it makes a cone with a definite slope, and pouring more makes a larger cone with the same slope. The reason the answer is an angle rather than a length is a cancellation — the force pulling a surface layer downhill and the friction holding it back are both proportional to its weight, so everything about the size of the pile divides out — and everything that puts a length back in is a story about cohesion.

fluids · Granular matter How much a wrap holds, against how many turns it is. The ratio of the two tensions a rope can hold across, against the number of turns it is wrapped, for coefficients of 0.1, 0.25, 0.5. The axis is logarithmic because the law is exponential, so each line is straight and its slope is the coefficient. At µ = 0.1 one turn multiplies by 1.9, two turns by 4 and three by 7 — so a person pulling with the strength of one arm holds a load that a small crane would be needed to lift. The practical consequence is the one a sailor states as a rule: turns are cheap and each is worth as much as the one before it, which is a statement about a constant factor rather than a constant force.

The part of the wrap that is actually gripping

The capstan equation gives the largest tension ratio a wrap can hold, and almost nothing spends its life at that limit. Below it the wrap divides in two: an idle arc doing nothing at all and an active arc creeping and carrying the whole exponential — and the division explains a belt's speed loss and its squeal.

mechanics · Friction The current crowding into a contact spot. A meridional section through a circular contact between two solids, with the spot at the centre. The closed curves are equipotentials and the curves running through the spot are current lines, each carrying an equal share. Both are exact: in the coordinates built on the spot's rim the equipotentials are confocal spheroids and the current lines confocal hyperboloids, and in this section they are ellipses and hyperbolas. What the picture shows is that the current has to converge from a region many spot radii across and then spread again, and that the potential falls almost entirely within a few radii of the contact. The equipotential drawn at 12% of the drop sits 5.2 radii away, and everything beyond it contributes that last 12%.

The resistance that is a length

Two metals touching do not touch over the area they appear to. Current crosses at a few small spots and has to converge into each one, and the resistance of that convergence contains no area and no path length — only the size of the spot, divided into the resistivity.

electromagnetism · Conductors The hourglass that keeps time. Discharge against how much is left above the opening, for grain and for liquid through the same 40 mm hole, each as a fraction of its own rate at a full hopper. The grain rate is a horizontal line: it does not appear in Beverloo's law at all, because the pressure at the outlet does not depend on the head — the walls carry the weight, which is what Janssen's argument establishes, so the grains at the opening are pushed by their immediate neighbours and by nothing else. The liquid falls as the square root of the head and is down to 32 per cent by the time a tenth is left. That is why an hourglass keeps time and a water clock does not, and why the water clocks that worked were built with a float and an overflow to hold the head constant.

The hourglass that keeps time

Grain leaves a hopper at a rate that does not depend on how much is above it, and that goes as the orifice to the five-halves power rather than the one half a liquid gives. Both facts follow from the same thing: the weight is carried by the walls, not by the grains at the opening.

fluids · Granular matter Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top.

The grip that needs a little slipping

A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable.

mechanics · Friction Grip, then power, then air. The force a 1500 kg car can put through its driven wheels against road speed, with 100 kW at the wheels and a tyre friction coefficient of 0.9. The grip allows 13.2 kN at any speed; the engine allows its power divided by the speed, a hyperbola; the car gets whichever is smaller, drawn solid. The two are equal at 7.6 m/s, 27 km/h: below it the car is limited by friction and extra power would change nothing, above it by power and better tyres would change nothing. The rising curve is the resistance, rolling plus air, which grows as the square of the speed; it meets the drive at 61.2 m/s, 220 km/h, the top speed, solved for and checked there.

The speed at which grip hands over to power

A car's specification lists its power, and power does not limit how hard a car can push. It limits how hard it can push at a given speed, and at low speed that limit is higher than anything the tyres can transmit. So every car leaves the line as a friction problem and becomes a power problem a second later, at a crossover speed that decides which upgrade would make it faster — and at the top of its speed range a third limit, the cube of the speed, takes over from both.

mechanics · Energy The pressure at which an oil becomes a glass. Viscosity against pressure for 4 liquids on a logarithmic axis, from Barus's rule with the pressure coefficient each one actually has. The rule is an exponential, so a gigapascal multiplies an ordinary oil's viscosity by a hundred million or more, and the curves cross the line conventionally taken to mark a glass — a million million pascal-seconds — at 1.37 GPa for a mineral oil, 0.98 GPa for a traction fluid. The vertical line is the peak pressure inside the loaded contact this figure is about, 1.32 gigapascals, computed from the Hertz solution for that geometry and load. Water is drawn for contrast: its pressure coefficient is thirty times smaller, it never approaches a glass over this range, and that is why it is useless as a lubricant in a rolling contact however clean it is. Each curve is drawn solid up to the glass line and dashed above it, because past that point the material is not a liquid and its viscosity is not what decides how it shears — the exponential continues, and the substance it was written for does not.

The oil that is a glass for a quarter of a millisecond

Every other account of viscosity here varies the temperature. Pressure does something larger and in the same direction for every liquid: viscosity rises exponentially, by a factor of ten to the eight or more at the gigapascal inside a loaded gear tooth. That is not a curiosity — it is the only reason there is a film there at all. Remove the pressure dependence from the calculation and the predicted film is five nanometres, under the roughness, and the surfaces touch.

fluids · Viscosity The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing.

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

mechanics · Energy Flow curves that bend back, and one that stops. Shear stress against shear rate, both on logarithmic axes, for volume fractions 0.5, 0.55, 0.565, 0.575, 0.6, from the same model: stress in units of σ, rate in units of σ over the liquid's viscosity. Below φ = 0.569 every curve rises steadily — the suspension thickens but continuously. Above it and below the frictional jamming point 0.58 the curve bends back into an S: over a range of stress, more stress gives less flow, and a suspension driven at a fixed rate must jump between branches. At φ = 0.6 the curve returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing altogether and holds the stress like a solid.

The suspension that seizes when pushed hard

Cornflour stirred into half its volume of water runs off a spoon like cream, and a fist driven into it bounces off as if it had hit a floor. The particles have not changed and neither has the water. What changes, above a threshold stress, is the kind of contact between neighbouring grains: lubricated, they slide past one another and the suspension can pack to 64 per cent before it locks; pressed into frictional contact, they lock at 58. A suspension sitting between those two numbers is a liquid when handled gently and a solid when handled hard.

fluids · Rheology The shear a pushed contact carries, and where it gives. Across a Hertzian contact of radius a, the normal pressure as a fraction of its peak, which is also the friction limit μp in units of μp₀ (dashed), and the shear traction for a sideways push of 0.3, 0.6, 0.9 times the load that would slide it, μP. Where the shear reaches the dashed ceiling the surfaces slip; inside, they stick. At 0.3 μP the stuck disc has a radius of 0.89a, so 21 per cent of the contact area is slipping; at 0.6 μP the stuck disc has a radius of 0.74a, so 46 per cent of the contact area is slipping; at 0.9 μP the stuck disc has a radius of 0.46a, so 78 per cent of the contact area is slipping. The shear is highest just inside the slipping ring and, in an elastic contact that simply stuck everywhere, would be infinite at the rim: slip at the edge is what removes that singularity.

The contact that slips before it slides

Static friction is supposed to hold perfectly until the load reaches μ times the normal force and then give way. Two elastic bodies pressed together do something else. Push a ball sideways on a flat with any force at all and a ring at the edge of their contact is already slipping, while the centre stays stuck. As the push grows the ring spreads inward, and gross sliding is only the moment the stuck centre shrinks to nothing. Shake the push back and forth and the ring slips each way on every cycle. The contact then loses energy though it never slides, as the cube of how hard it is shaken.

mechanics · Friction Every path a rolling ball can take on a turntable, seen from the room. The paths, seen from the room, of four solid balls rolling without slipping on a turntable of radius 15 cm turning at 33.3 revolutions a minute (grey disc), over 6.3 seconds. A ball set down at rest relative to the room stays exactly where it is, spinning on the spot while the table turns under it. Every other ball moves on a circle, turning the same way as the table at 2/7 of its rate — once every 3.5 turns of the table — and comes back to where it started, whatever the push. A ball set down moving with the table does not ride with it: it circles on a path three and a half times the radius it started at, and the circle's centre is on the other side of the axis.

The ball a turntable will not throw off

Set a marble down on a spinning turntable and the expectation is that it will be flung off the edge. It is not. Placed at rest relative to the room, it stays exactly where it is, spinning on the spot while the table turns beneath it. Given a push, it moves in a circle, turning the same way as the table at two-sevenths of the table's rate, and comes back to where it started. Tilt the turntable and the marble does not roll downhill: it wanders across the slope in a chain of loops. A rolling ball on a rotating surface obeys exactly the equation of an electric charge in a magnetic field, and the two-sevenths is how much of every push goes into turning the ball rather than moving it.

mechanics · Circular motion The pressure under a wheel rolling on a soft surface. The pressure under a rigid wheel of radius 30 cm carrying 3 kN per metre of width as it rolls over a rubber-like bed, against position along the contact (ahead of the axle to the right), scaled to the largest pressure and the widest contact, at three speeds. The bed relaxes with times of 2 and 20 ms. At 1 mm/s the pressure is symmetric and its resultant passes through the axle. At 1.0 m/s the material ahead is being squeezed faster than it can flow and pushes back harder than the recovering material behind, which lags and lets go early: the resultant sits 6.39 mm ahead of the axle. At 1000 m/s the bed has no time to relax at all and behaves as a stiffer elastic solid, symmetric again. A wheel is held back by where its support pushes, not by any sliding.

The push a millimetre ahead of the axle

A wheel rolling freely on a flat road slows down, and nothing in it slides. What holds it back is where the road pushes on it: not directly under the axle but a little ahead, because the material in front of the contact is being squeezed and pushes back harder than the material behind, which is recovering and lags. A car tyre's push is about three millimetres ahead of its axle, a train wheel's about two-thirds of a millimetre, and that offset, divided by the radius, is the whole of rolling resistance. It peaks at the speed where the time under the wheel matches the time the material takes to relax, and the same loss, at a different frequency, is what gives a tyre its grip.

mechanics · Friction Two ways for a swing to die. A mass on a spring released 10.6 dead-band widths from rest, against time in units of 1/ω. Red: sliding on a dry surface, whose friction is a constant force; each swing is exactly a cosine, and the turning points fall in a straight line by two dead-band widths per half-cycle until the mass stops, 2.5 cycles after release, at −0.60. Blue: the same mass with viscous damping (damping ratio 0.05), whose force is proportional to speed; the swings shrink by the same fraction each cycle, along an exponential, and never stop. The period is the same in both and the same as with no damping at all to within the viscous case's 0.13 per cent.

The swing that dies in a straight line

A pendulum in air loses the same fraction of its swing every cycle, so its amplitude falls along an exponential and, in principle, never quite reaches zero. A mass on a spring sliding over a dry surface does something else. Its friction is a fixed force, so it loses a fixed amount of amplitude every half-swing — the swings shrink in a straight line — and it stops completely, after a countable number of swings, somewhere inside a band around its natural position whose width is the friction divided by the stiffness. Where in that band it stops depends on where it started, which is why a barometer with friction in it is tapped before it is read.

mechanics · Friction The ladder that leaves the wall. A 4-metre ladder released from rest at 15° from the vertical, its foot on a frictionless floor and its top against a frictionless wall, drawn at equal intervals of time from release to the moment it lies flat, computed by integrating its motion; the dots trace its centre. While both ends touch, the centre moves exactly on a circle of radius L/2 about the corner (dashed). The top leaves the wall when it has come down to 2.58 m, two-thirds of its starting height of 3.86 m; from then on the centre's horizontal speed is fixed, it leaves the circle, and the ladder falls clear of the wall with its foot still sliding. The fall takes 1.34 seconds.

The ladder that lets go of the wall

A ladder whose foot slips on a smooth floor slides down a smooth wall — and partway down, its top comes away from the wall and the ladder falls the rest of the way clear of it. The point where it lets go of the wall is not set by its length, its mass or how steeply it started: it is always two-thirds of the height the top started at. The reason is a wall that can push but cannot pull, and the number falls out of energy alone, without a single force being calculated in advance.

mechanics · Energy A floor that has to hit a rod nobody dropped. The normal force on the lower end of a uniform rod at 63° with μ = 1.6, sliding outward at √(gl) and not yet turning, against time on a logarithmic axis, for a floor whose contact stiffness is 10⁴, 10⁵ and 10⁶ times mg/l. The rigid equations have no solution here, and the compliant floor shows why: the end digs in, the deeper it digs the harder friction twists it further down, and the force climbs until the sliding stops. The stiffer the floor, the higher and briefer the spike — 20, 56, 163 times the rod's weight, ending after 41, 18, 7 thousandths of √(l/g) — a collision with no approach.

The rod that friction cannot decide about

Newton's laws, a rigid rod and Coulomb's law of friction are the first things a mechanics course teaches, and in 1895 Paul Painlevé showed that together they can fail to give any answer at all. A rod sliding on a rough enough floor, at the right angle, has no motion consistent with the three — or has two. The way out is not a new law but a forgotten fact: real contacts give a little. With that, the rod jams, and the floor delivers a sudden blow to a rod that nothing struck — an impact without a collision, which is how a stick of chalk comes to hop.

mechanics · Friction How a layer of grains moves down a slope. The velocity through a steady layer of 0.5 mm glass beads 20 grains deep flowing down slopes of 22°, 25° and 28°, against height above the rough base in grain diameters; the dashed curve is a Newtonian liquid's parabola, scaled to the 25° surface speed. Because every depth must have friction tan θ, every depth has the same inertial number, and the shear rate grows with the square root of the pressure above it: Bagnold's profile, u ∝ h^(3/2) − (h − z)^(3/2). Its surface speeds are 8.0, 41.0, 113.7 cm/s; compared with a liquid's parabola the shear is spread more evenly through the depth, because the shear rate falls off only as the square root of the depth below the surface rather than in proportion to it.

The sand whose friction depends on how fast it flows

A heap of sand stands at its angle of repose and starts to slide a few degrees steeper; between those angles there are steady avalanches, and their speed is decided by one number. The inertial number compares how long a grain takes to fall into a gap opened under it with how long the flow takes to open the gap. Friction rises with it, from the static value to a ceiling, and that single dependence explains why grains flow steadily on only a twelve-degree window of slopes, why a flowing layer has the velocity profile Bagnold measured, and why a deeper avalanche is faster — but not as a liquid would be.

fluids · Granular matter

Named alongside it

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

Contact areaHysteresisNormal forceStatic frictionCoefficient of frictionContact mechanicsGranular matterScalingStick-slipContactDampingElasticity

All concepts