A ratio that is a derivative of a length

A block and tackle is the one mechanism whose ratio anybody can read off the picture. Count the parts of line running to the moving block; that is the ratio — no arithmetic, unlike a gear train. Four parts, four to one — and unlike a gear train there is no arithmetic, no tooth counts and nothing to look up.

The rule is exactly right, in the sense that it is the answer to a limit that a real tackle never quite reaches.

Where the rule comes from, and what it assumes #

A tackle is one strand. It has one length, and that length does not change, so whatever the mechanism does must leave the total alone. If the hauling end comes in by $\delta L$, the block separation must change by whatever amount takes $\delta L$ of strand out of the parts between the blocks.

If the parts are all parallel to the lift, each of them shortens by exactly the lift, so $n$ parts absorb $n$ times it and the ratio is $n$. That is the whole derivation, and the assumption is in the first clause.

The parts are not parallel. The sheaves in each block sit side by side, so the line running from one block to the other leaves at an angle, and a part at angle $\beta$ to the lift shortens by only $\cos\beta$ of it.

Differentiating the run instead #

The reason to build a tackle as a strand rather than as a count is that the strand has a length and a length can be differentiated. Move the blocks apart by a hundredth of a millimetre, ask the run how long it now is, and the difference divided by the movement is the velocity ratio — with no assumption anywhere about what a part of line is, how many there are, or which way any of them points.

For a four-part tackle with its sheaves 34 mm apart:

It approaches four and it reaches four nowhere, which is this site’s oldest theme in a new mechanism. And the case furthest from the rating is the one a tackle is actually used in: hauling a load the last few hundred millimetres, with the blocks close together, is where the mechanism does its work and where its ratio is 8% short of what it says on the shell.

The second route, which is the derivation done properly #

A ratio computed by finite differences is worth exactly as much as the second route that confirms it, and here the second route is the assumption above, repaired.

If a part at angle $\beta$ shortens by $\cos\beta$ of the lift, then the ratio is not $n$; it is $\sum \cos\beta_i$ over the parts that span the gap. Measure each part’s angle off the drawn run and add the cosines:

The two agree to every figure printed. That is the useful kind of agreement: the derivative knows nothing about parts of line, and the cosine sum knows nothing about lengths, and they are computed from different quantities of the same drawing.

It also says where the shortfall goes. At 120 mm the outer parts lean by 23°, and $\cos 23.37° = 0.918$: those two parts are each contributing less than 92% of a part. At 620 mm the worst lean is 4.6° and the cosine is 0.997.

What is actually in the length #

The run reports more than its total, and the breakdown is worth reading once because it says which part of the mechanism the ratio comes from.

At a separation of 220 mm the four-part tackle’s strand is 1,087.724 mm long: 918.252 mm of straight parts and 169.472 mm lying on the three sheaves. Pull the blocks out to 620 mm and the total is 2,677.033 — 2,519.473 straight and 157.560 wrapped. The straight portion nearly triples; the wrapped portion falls, by seven per cent.

That is the whole mechanism in two numbers. The wraps shrink because the parts are straightening: at close quarters a sheave carries 226.7° of line, at 620 mm it carries 189.1°, and a sheave with 180° of wrap is one whose two parts are parallel — which is the arrangement in which the count is the answer. The ratio and the wrap angles are measuring the same departure from parallel, and they approach their limits together.

None of the wrapped length participates in the ratio at all. Differentiate the run and the arcs contribute nothing to first order at the sheaves that stay put, because a wrap that grows at one end shrinks at the other. What moves the length is the straight parts, and it is exactly their leaning that the cosine sum counts.

Down at the limit of the travel #

The tackle gets worse faster than the plot’s right-hand end suggests, and the end of the travel is not off the page.

At 60 mm — with 34 mm between the sheaves, so the blocks are less than two spacings apart — the four-part tackle is delivering 3.21, a fifth short of its rating. Below that the blocks meet, which sailors call block and block and which is the position a tackle is in at the end of every haul.

So the number a tackle is sold with is its value at the beginning of the lift and its value at the end is materially different. Nothing about that is a defect; it is what reeving is. What is worth noticing is the direction: the mechanism is weakest exactly where the load has been lifted furthest, which is the opposite of the way a crank or a cam is usually arranged.

Velocity ratio, and the thing it is not #

One boundary, because the words are used interchangeably and this field can only compute one of them.

What is measured above is a velocity ratio: how much strand must be hauled per unit of lift. It is a statement about lengths and their derivatives, and every number in it survives with every force in the mechanism unknown.

The force ratio — what a tackle multiplies a pull by — is the reciprocal of the velocity ratio only in the absence of friction, and a tackle is a mechanism whose entire reputation rests on the difference. A six-part tackle passes its line over five sheaves; each of them takes a few per cent; the compounding is what decides whether the sixth part is worth having, and it is not a geometric question. This field says the four-part tackle gives 3.927 rather than 4 and stops there.

The count is not what decides the shortfall #

The obvious guess is that more parts means more spread means a bigger shortfall. Measured at one separation, that guess is wrong.

At 220 mm the two-part tackle is 2.33% short, the three-part 1.90%, the four-part 1.82%, the five-part 2.23% and the six-part 2.03%. The variation between them is smaller than the variation any one of them shows across its own travel, and it is not monotone in the count.

The reason is that adding a part does two things at once. It adds a leaning part to the sum, which hurts; and it spreads the existing sheaves differently, since the blocks are laid out symmetrically about the lift, which can help. A two-part tackle has both its sheaves off-centre by half a spacing; a three-part tackle has one of them dead on the axis, contributing $\cos 0°$ exactly.

So the design variable is not the number of parts. It is the ratio of sheave spacing to working separation, and a tackle used at close quarters is compromised whatever its reeving.

Which block carries the dead end #

One structural fact falls out of building the mechanism rather than counting it, and it is easy to get wrong on paper.

The strand alternates between the two blocks. So if the dead end is made off to the standing block, the parts spanning the gap come in pairs, and an odd number of them is impossible: the last span would run from a block back to itself, supporting nothing. An even tackle ends its line on the standing block and an odd one on the moving block.

Get it wrong and the mechanism still assembles, still draws, and quietly delivers one part fewer than it was designed for. A three-part reeving with the dead end above measures a ratio of 1.973 — a two-part tackle with an extra sheave in it. Nothing about the drawing looks wrong; the extra span is there on the page, running from the standing block to the standing block, taking its share of the line and none of the load.

The same derivative, everywhere else #

The tackle is the clearest case, and the technique is the field’s rather than the tackle’s: a strand’s ratio is the derivative of its length with respect to whatever is moving.

Differentiate with respect to a body’s position and the answer is a velocity ratio, as here. Differentiate with respect to a body’s rotation and the answer is an arm — the perpendicular distance from the axis to the strand, which for a round pulley turning about its own centre is its radius and in general is not.

Both are computed the same way and both are checked the same way — against a closed form that is correct for the arrangement it was derived for. That is worth insisting on, because every one of the closed forms in this subject is right: the tackle’s $n$, the tendon’s $r$, the variator’s belt-length relation. They are right about the arrangement they assume and silent about how much they lose when it is not exactly that, and the loss is what a designer needs.

One strand, one equation, and no turning number #

It is worth saying plainly that the whole tackle is one run. The dead end, three sheaves and the hauling end are five stations in one list, and the length that gets differentiated is the length of all of it — including the part above the standing block, which does not move and contributes nothing.

That is the same one-scalar-equation structure the rest of this field has, with one difference: a tackle’s run is open. It starts somewhere and ends somewhere else, so it does not close, and none of the arithmetic about turning numbers applies to it. There is no integer to check against. What replaces it here is the cosine sum, which is a genuinely independent route rather than an invariant, and this is the only mechanism in the field where the two are different in kind.

An open run also has no constraint of its own until something holds both ends. The tackle’s length is fixed because the dead end is made off and the hauling end is held; let go of the hauling end and the length constraint stops being active, the mechanism has a freedom back, and the load falls. That is the unilateral constraint of this field seen at its most literal, and it is the subject of the essay on slack.

Where the hauling end goes #

A powered tackle winds its hauling part onto a drum, and the two geometries then compound.

The tackle contributes a ratio between 3.2 and 4.0 depending on where in its travel it is. The drum contributes a line speed per turn of $2\pi r$, and $r$ is the radius of whichever layer is being wound rather than a property of the drum: on the winch in the essay on drums it runs from 207.35 mm per turn on the first layer to 395.84 on the sixth, a factor of 1.909.

Multiplied together, a winch and tackle whose nameplate says one number is running at a lift speed that varies by a factor of about two over its working stroke, from two causes that have nothing to do with each other and both of which are pure geometry.

The difference mechanisms, which are the opposite case #

A tackle multiplies by counting. The other family of strand hoists multiplies by subtracting: a differential chain hoist has two sprockets of nearly equal radius on one shaft, and its ratio is $2R/(R - r)$, which grows without bound as the two are brought together.

That mechanism is already on this site, met from the gear train’s side, and what it shares with a differential screw and a compound epicyclic is a conditioning number of $|a/(a - b)|$ — a ratio that is enormous precisely because it is the difference of two nearly equal things, and therefore enormously sensitive to both of them.

A tackle has none of that. Its ratio is a sum rather than a difference, its conditioning is perfect, and its shortfall is a smooth geometric 2% rather than a sensitivity. The two families are worth putting side by side because they are the only two ways a strand multiplies, and they fail in opposite ways: one is exact and fragile, the other approximate and robust.

The shortfall is second order in the lean #

The mean of cosines is the exact expression for the ratio, and expanding it says how the shortfall behaves rather than merely what it is at two separations — which is what turns the measurement into a design rule.

A part leaning at $\beta$ contributes $\cos\beta \approx 1 - \beta^2/2$, so its shortfall against unity is second order in its lean. The tackle’s ratio is the sum of those, so the whole shortfall is a mean of $\beta_i^2/2$: small angles cost almost nothing, and the cost grows as the square rather than in proportion.

Put the numbers to it. At 220 mm the worst part leans 12.76°, which is 0.223 radians, and half its square is 2.5 per cent — an upper bound on the shortfall, since the other parts lean less, against a measured 1.82 per cent. The arithmetic and the measurement agree in the way a bound and a mean should.

Inverted, that is the rule a rigger would want. Keep every part within about ten degrees of the lift and the tackle is within one and a half per cent of its count. Ten degrees is 0.175 radians, half its square is 1.5 per cent, and no part can be worse than the worst one. That is a criterion on an angle, checkable by eye on the hardware, and it does not need the block separation or the sheave spacing to be measured.

It also says exactly which geometry to change, because the lean is set by the ratio of the sheave spacing to the block separation and by nothing else. Halving the sheave spacing halves every lean and quarters the shortfall; doubling the separation does the same. So a tackle that is short is short because its blocks are close relative to how wide they are, and the two remedies are a narrower block or a longer lift.

Which explains the shape of real tackle. Blocks are made narrow — sheaves close together, side by side on one pin — and the reason is usually given as compactness. The geometry says it is also the whole of the ratio’s accuracy: a wide block is a tackle whose parts lean, and a tackle whose parts lean does not give the number it is named for anywhere in its travel.

What is not modelled #

There is no friction anywhere in this, which means the shortfall computed here is not the reason a real tackle underperforms. A sheave on a plain bearing loses several per cent per sheave, and a six-part tackle passes its line over five of them; the friction loss is the dominant term by a long way and it needs a bearing, a load and a coefficient. What is measured here is the part of the shortfall that survives with every force unknown, and it is worth separating for exactly that reason: it is present in a frictionless tackle, it depends only on the geometry, and no amount of lubrication removes it.

Nothing here models the rope’s thickness, its stiffness against bending round a sheave, or the fact that a real rope’s parts are not free to be at whatever angle the geometry wants. The blocks are rigid and remain parallel, which a real pair does not. And the load hangs from the moving block’s centre, so nothing tilts.