Which of the following are correct about Merchant’s relation `2φ + β − α = 90°`?

It is derived by minimising the cutting energy with respect to the shear angle

It predicts that increasing the rake angle increases the shear angle

Measured shear angles are generally smaller than it predicts

It is derived by minimising the cutting energy with respect to the shear angle

It predicts that increasing the rake angle increases the shear angle

Measured shear angles are generally smaller than it predicts

It holds exactly for any real cut with a continuous chip

According to Taylor’s principle of gauge design, a NO-GO plug gauge for a hole should be

made to the maximum material limit and the full length of engagement

made to the mean size and the full length of engagement

made to the minimum material limit and the full length of engagement

Why is the hole-basis system preferred over the shaft-basis system for most fits?

A hole is produced by a fixed-size tool, while a shaft can be turned to any size on the same machine

Holes can be measured more accurately than shafts

A hole is produced by a fixed-size tool, while a shaft can be turned to any size on the same machine

Holes are always in tension and shafts in compression

The hole-basis system gives tighter tolerances

A CNC program written in incremental mode (`G91`) rather than absolute mode (`G90`) is riskier mainly because

a single coordinate error displaces every subsequent move

incremental mode cannot do circular interpolation

a single coordinate error displaces every subsequent move

incremental mode runs the spindle in the wrong direction

incremental mode does not support tool-offset compensation

A factory makes 300 different parts in medium volumes. It reorganises its machines from a process layout into cells, each cell handling one part family. The largest single benefit is

a large reduction in setup time, which lowers the economic batch size and so the work-in-progress

a higher cutting speed on each machine

the ability to make parts the process layout could not make at all

elimination of the need for CNC machines

Machining, Metrology and Computer Integrated Manufacturing

Three sub-items sit here and the first carries most of the marks. Machining is analysed through the orthogonal model, whose one job is to turn a measurable chip into an unmeasurable shear plane: the chip-thickness ratio r = t/t_c gives the shear angle through tan φ = r cos α/(1 − r sin α), and from φ everything else follows — shear strain, force components, specific energy. Two things about that model deserve saying at the start. It is a plane-strain idealisation with a single continuous chip and no built-up edge, so it describes a lathe cut reasonably and a milling cut only loosely. And Merchant’s relation 2φ + β − α = 90° is a minimum-energy prediction, not a measurement, and real chips are consistently thicker than it says — the gap is not an error in your arithmetic. Tool life is Taylor’s VT^n = C, and the honest way to read it is as an exponent problem: with n = 0.25, doubling the cutting speed does not halve tool life, it divides it by sixteen, and that single non-linearity is why machining economics is a real optimisation rather than a preference. Metrology is about tolerance rather than measurement — limits and fits, and Taylor’s principle, which says a GO gauge must check every dimension at once and a NO-GO gauge must check one at a time. And CIM is the vocabulary layer: NC and CNC, part programming, group technology, FMS, and the reason a flexible cell exists at all is a batch size too large for a job shop and too small for a transfer line.

From a Chip You Can Measure to a Shear Plane You Cannot

The orthogonal model reduces cutting to a single shear plane running from the tool tip to the free surface, inclined at the shear angle φ. You cannot measure φ directly, but you can measure the chip: weigh it, or measure its thickness, and the chip-thickness ratio r = t/t_c follows, where t is the uncut thickness (the feed) and t_c the chip thickness. Then tan φ = r cos α/(1 − r sin α), with α the rake angle. Work one. Uncut thickness 0.2 mm, chip thickness 0.4 mm, rake 10°: r = 0.5, so tan φ = 0.5 cos 10°/(1 − 0.5 sin 10°) = 0.4924/0.9132 = 0.5392 and φ = 28.33°. From that, the shear strain the metal suffers is γ = cot φ + tan(φ − α) = 1.856 + 0.331 = 2.186 — over two hundred per cent strain, in a zone a few hundredths of a millimetre thick, which is why cutting temperatures are what they are. Notice what the chip being thicker than the feed means physically: the chip is shorter and fatter than the metal it came from, which is compression, and a larger r — a thinner chip — means a larger φ, a shorter shear plane, less work and a cooler cut. That is the whole reason positive rake angles and sharp tools are worth having.

⚠️ Merchant’s relation is a prediction, and real chips disagree with it

Merchant’s relation 2φ + β − α = 90° comes from minimising the cutting energy with respect to φ, and it is worth being clear that this makes it a model’s prediction of what the shear angle ought to be, not a measurement of what it is. Run both directions on the same numbers and the gap is visible. For α = 10° and a friction angle β = 30°, Merchant predicts φ = 45 + α/2 − β/2 = 35°. But the chip measured above — 0.2 mm feed giving a 0.4 mm chip at the same 10° rake — has φ = 28.33°, and feeding that back through Merchant’s relation implies β = 43.33° rather than 30. Real chips are consistently thicker, and real shear angles consistently smaller, than the minimum-energy prediction. The discrepancy is not arithmetic; it is the model. Merchant assumes a single thin shear plane, a constant coefficient of friction on the rake face, no built-up edge, no strain-hardening in the shear zone and no work done in the secondary zone where the chip rubs along the tool — and every one of those assumptions fails to some degree in a real cut. So use the relation as an examiners use it: to get β from a measured φ, or to predict how φ moves when α changes, which it does correctly. Do not use it to compute an absolute shear angle and then treat a disagreeing measurement as an error. The practical direction it gets right is the useful one: increasing rake angle increases the shear angle, which thins the chip, shortens the shear plane and cuts the force — which is exactly why tool designers spend geometry on rake.

Taylor’s tool life `V T^n = C` with n = 0.25 and C = 200
Cutting speed	Tool life	Relative to 100 m/min
50 m/min	256 min	16× longer
100 m/min	16 min	1× — the reference
200 m/min	1 min	1/16 as long
400 m/min	3.75 seconds	1/256 as long

🎯 Why `n = 0.25` makes machining economics a real optimisation

Taylor’s law is V T^n = C, so T = (C/V)^(1/n) — and with the exponent inverted, a small n makes tool life ferociously sensitive to speed. Fix C from one experiment: at V = 100 m/min a tool lasts 16 minutes, so C = 100 × 16^0.25 = 200 exactly. Now read the consequences. Doubling the speed to 200 m/min drops tool life to 1 minute — a factor of 2^(1/n) = 2⁴ = 16, not 2. Halving it to 50 m/min stretches life to 256 minutes. A three-second change in the sums produces a sixteen-fold change in tooling cost, and that is why cutting speed is chosen by optimisation rather than by preference. The trade is explicit. Higher speed means less machining time per part but more tool changes, more inserts consumed and more idle time regrinding; lower speed means the opposite. Minimising cost per part gives one optimum speed, and minimising production time per part gives a different, higher one — the two do not coincide, and which you want depends on whether the machine or the tooling is the scarce resource. Two honest limits on the law. n is a property of the tool material, roughly 0.1 for high-speed steel, 0.2–0.3 for carbide and up to 0.5 for ceramics, so a carbide tool tolerates a speed increase far better than HSS — that spread is the reason tool material selection changes the whole speed regime and not just the wear rate. And C shifts with feed, depth of cut, workpiece material and coolant, so a C measured in one setup does not transfer to another.

Metrology: Tolerance Is the Subject, Not Measurement

A fit is the relationship between the tolerance zones of a hole and a shaft, and the three families are decided by whether those zones overlap. A clearance fit has the smallest hole larger than the largest shaft, so the parts always assemble and always have play; an interference fit has the largest hole smaller than the smallest shaft, so assembly needs force or heat and the joint transmits torque by friction alone; a transition fit has zones that overlap, so a particular pair may come out either way. The two conventions matter because they answer different manufacturing questions. In the hole basis system, the hole is held at one tolerance and the shaft is varied — the standard choice, because a hole is made by a fixed-size tool (a reamer, a broach) and varying it means buying more tools, whereas a shaft can be turned to any size on the same machine. Shaft basis is used where a single drawn bar of fixed size must fit several different housings, as in a bearing on a long shaft. Read a fit specification as three separate facts: the letter gives the position of the tolerance zone relative to the basic size, the number gives its grade (width), and capitals are holes while lower case are shafts. So H7/g6 is a hole whose lower limit is exactly the basic size, paired with a shaft sitting just below it — a close running fit.

🧠 Taylor’s principle: GO checks everything at once, NO-GO checks one thing at a time

Gauge design has one governing rule and it is asymmetric, which is exactly what makes it examinable. The GO gauge must check all the dimensions of a feature simultaneously, so it is made to the maximum material limit and to the full length of engagement. The NO-GO gauge must check one dimension at a time, so it is made to the minimum material limit and kept short. Work through why. A GO plug gauge for a hole is a full-length cylinder at the smallest permitted hole size: if it enters, the hole is everywhere at least that big and straight enough and round enough over the whole depth — one insertion has verified size, roundness and straightness together, which is precisely what an assembly needs, because a mating shaft also has to pass through the whole length. Now the NO-GO. It is made to the largest permitted size and deliberately made short, so it can only sample one cross-section: if it were full length, a hole that was slightly oversize at one spot but curved would refuse the gauge and be passed as good, and the defect would go undetected. Short and single-purpose is what lets it catch the local error. The same asymmetry runs through ring and snap gauges for shafts, with the limits swapped. And the reason a NO-GO gauge is often made as a snap rather than a full ring is exactly this: a snap contacts two points at one axial position, which is the minimum needed to measure a diameter and nothing more.

CIM: The Cell Exists Because of Batch Size

Computer integrated manufacturing is mostly vocabulary, and the vocabulary is easiest to hold if you keep asking what problem each term solves. NC replaced a machinist’s handwheels with coded instructions on tape; CNC put a computer at the machine so the program could be edited, stored and run with tool-offset compensation; DNC networked several of them to a host. Part programming is then the interface: G-codes command motion and mode — G00 rapid, G01 linear feed, G02/G03 circular interpolation clockwise and anticlockwise, G90/G91 absolute and incremental — while M-codes handle the machine functions, spindle, coolant and tool change. The distinction worth being firm about is absolute against incremental: in G90 every coordinate is measured from one fixed origin, so a single arithmetic slip affects one move, while in G91 each move is measured from where the tool currently is, so one slip displaces every subsequent move. That error-propagation difference is the practical reason absolute programming is the default for anything long.

🎯 Group technology and FMS answer a question about volume, not about technology

Two classical layouts sit at the extremes. A job shop groups machines by type — all the lathes together, all the mills together — which is flexible and has terrible material flow, because every part wanders across the floor and spends most of its life queueing. A transfer line arranges machines in the sequence one product needs, which gives superb flow and no flexibility at all, because a design change means rebuilding the line. Neither suits the medium volume in between, and that gap is the entire reason the middle ground was invented. Group technology attacks it by classifying parts into families that need similar operations, then arranging machines into cells, each cell owning one family. Each cell then behaves like a small transfer line for its family and like a job shop for the variety within it, which is exactly the compromise medium volume needs. An FMS is that cell automated — CNC machines linked by automated material handling, usually AGVs or a shuttle, under one computer that schedules the work. The payoff is worth naming precisely: setup time collapses, because a cell’s machines are already tooled for its family, and setup is the cost that makes small batches uneconomic in a job shop. That is the connection back to inventory: reducing setup cost reduces the economic batch size, which reduces work-in-progress, which is where most of a factory’s tied-up capital sits. So group technology is not a machine-layout idea that happens to help inventory; it is an inventory idea implemented as a layout, and questions that ask why a cell was installed are usually asking for that chain.

Key takeaways

Measure the chip, get the shear plane: tan φ = r cos α/(1 − r sin α). At t = 0.2, t_c = 0.4, α = 10°: r = 0.5, φ = 28.33°, shear strain γ = 2.186.
Merchant’s 2φ + β − α = 90° is a minimum-energy prediction, not a measurement. It predicts 35° for α = 10°, β = 30° while the measured chip above gives 28.33°. Use it for direction of change and to back out β, not for an absolute φ.
Taylor is an exponent problem. With n = 0.25 and C = 200, tool life is 256 min at 50 m/min, 16 at 100, 1 at 200 — doubling speed divides life by 2^(1/n) = 16.
n is a property of the tool material — about 0.1 for HSS, 0.2–0.3 for carbide, up to 0.5 for ceramics — so tool choice changes the whole speed regime, not just the wear rate.
Hole basis is standard because a hole is made by a fixed-size tool. Capitals are holes, lower case shafts; the letter positions the tolerance zone and the number grades its width.
Taylor’s gauge principle is asymmetric: GO is full-length at the maximum material limit and checks size, roundness and straightness at once; NO-GO is short at the minimum material limit and checks one section at a time.
A cell exists because of batch size. Job shop = flexible with terrible flow; transfer line = superb flow with no flexibility; group technology and FMS fill the medium-volume gap by collapsing setup time, which is an inventory result implemented as a layout.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

In orthogonal cutting with a rake angle of 10°, the uncut chip thickness is 0.2 mm and the chip thickness is 0.4 mm. The shear angle, in degrees, is ______ (round to two decimals)
Numerical answer — type the value.
Show answer
Answer: 28.33
28.33°. The chip-thickness ratio is r = t/t_c = 0.2/0.4 = 0.5, and tan φ = r cos α/(1 − r sin α) = 0.5 × 0.9848/(1 − 0.5 × 0.1736) = 0.4924/0.9132 = 0.5392, so φ = 28.334°. From that the shear strain follows, γ = cot φ + tan(φ − α) = 1.856 + 0.331 = 2.186 — over two hundred per cent strain concentrated in a zone a few hundredths of a millimetre thick, which is why cutting is a high-temperature process however cool the workpiece looks. Two checks worth carrying. r is always less than 1 for a real cut, because the chip is shorter and thicker than the metal it came from; if your r exceeds 1 you have divided the wrong way. And the trend matters more than the number: a thinner chip means a larger r, hence a larger φ, a shorter shear plane, less work and a cooler cut — which is precisely why positive rake angles and sharp tools pay for themselves. Note what this answer is not: Merchant’s relation would predict φ = 45 + α/2 − β/2 = 35° for a friction angle of 30°, and the 28.33° measured here is genuinely smaller. Real shear angles come out below the minimum-energy prediction, and that gap is the model’s limitation rather than an arithmetic slip.
A tool lasts 16 minutes at a cutting speed of 100 m/min. Taylor’s exponent is n = 0.25. The tool life at 200 m/min, in minutes, is ______
Numerical answer — type the value.
Show answer
Answer: 1
1 minute. Fix the constant first: C = V T^n = 100 × 16^0.25 = 100 × 2 = 200 exactly. Then at V = 200, T = (C/V)^(1/n) = (200/200)^4 = 1 minute. The striking part is the factor: doubling the speed divided tool life by sixteen, not by two, because T goes as V^(−1/n) and 1/n = 4. Run it the other way as a check — halving the speed to 50 m/min gives T = (200/50)^4 = 4⁴ = 256 minutes, again a factor of sixteen. This non-linearity is why cutting speed is chosen by optimisation and not by preference: raising the speed cuts machining time per part but multiplies tool consumption and tool-change downtime, and the speed that minimises cost per part is lower than the speed that minimises time per part, so the two optima genuinely differ and which one you want depends on whether tooling or machine capacity is scarce. Two honest limits. n is a property of the tool material — roughly 0.1 for high-speed steel, 0.2 to 0.3 for carbide, up to 0.5 for ceramics — so a carbide tool tolerates a speed increase far better than HSS, and switching tool material changes the whole speed regime rather than just the wear rate. And C is not universal: it shifts with feed, depth of cut, workpiece material and coolant, so a constant measured in one setup does not carry to another.
Which of the following are correct about Merchant’s relation 2φ + β − α = 90°?
1. It is derived by minimising the cutting energy with respect to the shear angle
2. It predicts that increasing the rake angle increases the shear angle
3. Measured shear angles are generally smaller than it predicts
4. It holds exactly for any real cut with a continuous chip
Show answer
Answer: A — It is derived by minimising the cutting energy with respect to the shear angle; B — It predicts that increasing the rake angle increases the shear angle; C — Measured shear angles are generally smaller than it predicts
A, B and C. A is the derivation: Merchant assumed the shear angle takes whatever value minimises the total cutting energy, differentiated, and got the relation. B follows from the algebra — rearranged, φ = 45° + α/2 − β/2, so φ rises with α, which is the correct and practically useful prediction: a larger rake thins the chip, shortens the shear plane and cuts the force, which is why tool geometry spends so much on rake. C is the empirical fact and the reason the relation must be handled carefully: real chips are consistently thicker and real shear angles consistently smaller than predicted. For α = 10° and β = 30°, Merchant gives 35°, while the worked chip in this chapter — 0.2 mm feed, 0.4 mm chip, same rake — gives 28.33°. D is therefore false, and being clear about why matters more than the answer: Merchant assumes a single thin shear plane, a constant coefficient of friction over the rake face, no built-up edge, no strain hardening in the shear zone and no work in the secondary zone where the chip rubs along the tool. Every one of those fails to some degree in a real cut, and the discrepancy is the model rather than the measurement. The working rule is to use the relation for the direction of a change and to back out β from a measured φ, never to compute an absolute shear angle and then treat a disagreeing chip as an error.
According to Taylor’s principle of gauge design, a NO-GO plug gauge for a hole should be
1. made to the maximum material limit and the full length of engagement
2. made to the minimum material limit and kept short
3. made to the mean size and the full length of engagement
4. made to the minimum material limit and the full length of engagement
Show answer
Answer: B — made to the minimum material limit and kept short
Minimum material limit, kept short. Taylor’s principle is deliberately asymmetric, and the asymmetry is the whole content. The GO gauge must check every dimension of the feature simultaneously, so it is made to the maximum material limit — for a hole, the smallest permitted size — and to the full length of engagement: if it enters, the hole is everywhere at least that big, straight enough and round enough over the whole depth, which is exactly what a mating shaft will demand. That is option A, and it is the GO gauge, not the NO-GO. The NO-GO gauge must check one dimension at a time, so it is made to the minimum material limit — the largest permitted hole — and made short on purpose, so it can only sample one cross-section. Option D is the trap and it is worth seeing exactly what it breaks: a full-length NO-GO gauge on a hole that is oversize at one spot but slightly curved would refuse to enter, the part would be passed as good, and the local defect would escape. Short and single-purpose is what lets it catch a local error. Option C would tell you nothing, since a gauge at the mean size answers no question about either limit. The same asymmetry runs through ring and snap gauges for shafts with the limits swapped, and it is why a NO-GO for a shaft is usually a snap rather than a ring: a snap contacts two points at one axial station, which is the minimum needed to measure a diameter and nothing more.
Why is the hole-basis system preferred over the shaft-basis system for most fits?
1. Holes can be measured more accurately than shafts
2. A hole is produced by a fixed-size tool, while a shaft can be turned to any size on the same machine
3. Holes are always in tension and shafts in compression
4. The hole-basis system gives tighter tolerances
Show answer
Answer: B — A hole is produced by a fixed-size tool, while a shaft can be turned to any size on the same machine
Because a hole is made by a fixed-size tool. Reamers, broaches and drills come in discrete sizes, so varying the hole diameter across a family of fits means buying and storing a separate tool for each — whereas a shaft is turned or ground and the operator simply takes a different final cut on the same machine with the same tool. Holding the hole at one tolerance (an H grade, whose lower limit is the basic size) and varying the shaft therefore minimises tooling investment, which is the entire economic argument. Option A inverts the metrology: holes are harder to measure accurately than shafts, because internal measurement is awkward and the gauge cannot be applied from outside, which is a reason to standardise them rather than to prefer them for measurement. Option C is not a real distinction — a fit says nothing about the stress state, and an interference fit puts the hole in tension and the shaft in compression only when it has been assembled. Option D confuses the two systems’ purpose: both use the same tolerance grades, so neither gives a tighter fit than the other; they differ only in which member is held constant. And shaft basis does have its place: where a single drawn or ground bar of fixed size must fit several different housings — a long transmission shaft carrying bearings, pulleys and couplings — it is the shaft that cannot be varied, and holding it constant while varying the holes is then the cheaper choice.
A CNC program written in incremental mode (G91) rather than absolute mode (G90) is riskier mainly because
1. incremental mode cannot do circular interpolation
2. a single coordinate error displaces every subsequent move
3. incremental mode runs the spindle in the wrong direction
4. incremental mode does not support tool-offset compensation
Show answer
Answer: B — a single coordinate error displaces every subsequent move
A single error displaces every subsequent move. In absolute mode (G90) each coordinate is measured from one fixed program origin, so a mistyped value puts exactly one move in the wrong place and everything after it is still referenced correctly. In incremental mode (G91) each coordinate is a displacement from wherever the tool currently is, so an error shifts the tool’s position permanently and every move after it inherits the offset — a 1 mm slip early in a hundred-line program leaves the last ninety-nine features all 1 mm out. That error-propagation asymmetry is the practical reason absolute programming is the default for anything long, and why incremental mode is reserved for repeated identical patterns, such as a bolt circle or a row of holes, where the displacement really is the natural way to express the geometry. Options A, C and D are all fabrications: incremental mode does circular interpolation with G02 and G03 exactly as absolute does, has nothing to do with spindle direction — that is an M-code function — and supports tool-offset compensation identically, since offsets are applied by the control after the coordinates are resolved. Worth keeping the code families separate as well: G-codes command motion and mode (G00 rapid, G01 linear feed, G02/G03 circular, G90/G91 the two modes here), while M-codes command machine functions — spindle on and off, coolant, tool change, program end.
Taylor’s exponent for a tool is n = 0.25. If the cutting speed is increased by 20 per cent, the tool life falls by ______ per cent (round to two decimals)
Numerical answer — type the value.
Show answer
Answer: 51.77
51.77 per cent. Taylor’s law rearranged is T ∝ V^(−1/n) = V^(−4), so multiplying the speed by 1.2 multiplies the life by 1.2^(−4) = 1/2.0736 = 0.4823 — the tool now lasts 48.23 per cent of what it did, a drop of 100 − 48.23 = 51.77 per cent. A 20 per cent speed increase more than halves tool life, and stating it that way is the most useful form of the result: the speed increase that exactly halves tool life is only 2^0.25 − 1 = 18.92 per cent. Two habits follow from this. First, work these problems as ratios, never by finding C and then two absolute lives — the ratio form T₂/T₁ = (V₁/V₂)^(1/n) needs no constant at all and cannot go wrong on units. Second, be sure which exponent is inverted: the law is V T^n = C with n small, so T carries the large exponent 1/n, and answering with 1.2^(−0.25) = 0.9646 — a 3.5 per cent drop — is the standard error and is out by a factor of fifteen. The engineering consequence is why cutting speed is optimised rather than chosen: a modest speed rise buys a modest cut in machining time and pays for it with a large rise in tool consumption and tool-change downtime, so the speed minimising cost per part sits well below the speed minimising time per part.
A factory makes 300 different parts in medium volumes. It reorganises its machines from a process layout into cells, each cell handling one part family. The largest single benefit is
1. a large reduction in setup time, which lowers the economic batch size and so the work-in-progress
2. a higher cutting speed on each machine
3. the ability to make parts the process layout could not make at all
4. elimination of the need for CNC machines
Show answer
Answer: A — a large reduction in setup time, which lowers the economic batch size and so the work-in-progress
Setup-time reduction, and its consequences downstream. In a process layout each machine sees an arbitrary sequence of unrelated parts, so almost every job needs a fresh setup — new fixtures, new tooling, new offsets. Group the parts into families that need similar operations and dedicate a cell to each, and the machines in that cell are already tooled for what arrives, so changeovers become minor. The chain that follows is where the real payoff sits: setup cost is what makes small batches uneconomic, so cutting it lowers the economic batch size, which lowers work-in-progress inventory, which is where most of a factory’s tied-up capital and most of its throughput time live. Add shorter travel distances and simpler scheduling and the cell wins on flow as well. Option B confuses layout with process parameters — cutting speed is set by tool material, workpiece and Taylor economics, and rearranging machines does not change it. Option C overclaims: a process layout can make every one of those parts, just slowly and with high inventory; the cell changes the economics, not the capability. Option D has it backwards — an FMS is precisely a cell whose machines are CNC, linked by automated handling under one scheduler, and CNC is what makes a cell able to absorb the variety within its family. The framing worth carrying is that group technology is not a layout idea that happens to help inventory; it is an inventory result implemented as a layout, aimed squarely at the medium volumes that are too large for a job shop and too small for a transfer line.