Generator

What friction returns, against what it is asked for

One function in the mechanics library, called 56 times across 9 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws 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.

friction-response is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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 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.

Which failure comes first

The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

A block on a 22° incline

The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

A block on a 22° 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.

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.

Where the floor is actually pushing back

The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Where the floor is actually pushing back. A block 300 mm wide and 900 mm tall, pushed horizontally at 100% of its height, on a surface with μ_s = 0.42. The curve is the position of the resultant of the normal force — the single place the whole distributed reaction of the floor could be replaced by — as a fraction of the way from the centre of the base to its leading edge. It walks outward in proportion to the push, because moment balance puts it at P·y/W, and it reaches the edge at P/W = 0.167. Friction gives way at P/W = 0.420. Whichever comes first is what happens, and here the block tips. Nothing about the block's mass appears anywhere: both thresholds are fractions of its weight.

A block 300 mm wide and 900 mm tall, pushed horizontally at 100% of its height, on a surface with μ_s = 0.42. The curve is the position of the resultant of the normal force — the single place the whole distributed reaction of the floor could be replaced by — as a fraction of the way from the centre of the base to its leading edge. It walks outward in proportion to the push, because moment balance puts it at P·y/W, and it reaches the edge at P/W = 0.167. Friction gives way at P/W = 0.420. Whichever comes first is what happens, and here the block tips. Nothing about the block's mass appears anywhere: both thresholds are fractions of its weight.

Where the floor is actually pushing back

The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Where the floor is actually pushing back. A block 300 mm wide and 900 mm tall, pushed horizontally at 16% of its height, on a surface with μ_s = 0.42. The curve is the position of the resultant of the normal force — the single place the whole distributed reaction of the floor could be replaced by — as a fraction of the way from the centre of the base to its leading edge. It walks outward in proportion to the push, because moment balance puts it at P·y/W, and it reaches the edge at P/W = 1.042. Friction gives way at P/W = 0.420. Whichever comes first is what happens, and here the block slides. Nothing about the block's mass appears anywhere: both thresholds are fractions of its weight.

A block 300 mm wide and 900 mm tall, pushed horizontally at 16% of its height, on a surface with μ_s = 0.42. The curve is the position of the resultant of the normal force — the single place the whole distributed reaction of the floor could be replaced by — as a fraction of the way from the centre of the base to its leading edge. It walks outward in proportion to the push, because moment balance puts it at P·y/W, and it reaches the edge at P/W = 1.042. Friction gives way at P/W = 0.420. Whichever comes first is what happens, and here the block slides. Nothing about the block's mass appears anywhere: both thresholds are fractions of its weight.

Which failure comes first

The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a wine glass, a stacked chair, a wardrobe on carpet.

The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a wine glass, a stacked chair, a wardrobe on carpet.

What checks it

physicscheck asserts something about friction-response that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Slide or topple

Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.

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.

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.

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.

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.

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.

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.

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.

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.

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