Form, Surface and Machine-Element Metrology

Introduction

Geometric accuracy determines whether a manufactured component will assemble, move, seal, transmit power, or survive its intended service life. A shaft may have the correct diameter yet fail because it is bent, lobed, rough, or eccentric. A thread may have the correct nominal size but not assemble because its pitch diameter or flank angle is incorrect. Metrology of form, surface texture, and machine elements therefore extends dimensional inspection from simple size measurement to the complete geometric and functional condition of a part.

This chapter deals with the measurement and interpretation of straightness, flatness, roundness, cylindricity, runout, surface texture, threads, gears, tapers, and related machine elements. The central principle is that every measurement must be related to a specification, a datum or reference system, a suitable instrument, and a stated measurement uncertainty. Standards such as ISO GPS standards, ISO 4287/4288 and ISO 21920 for surface texture, ISO 1101 for geometrical tolerancing, ISO 5459 for datums, ISO 12233 for threads, ISO 1328 for gears, and applicable ASME Y14.5 and ASME B-series standards provide the language and acceptance framework. The exact edition and national adoption must always be verified before inspection.

Learning objectives

After studying this chapter, the learner should be able to:

1. Distinguish size, form, orientation, location, and runout errors.
2. Explain reference elements and the minimum-zone principle used in form evaluation.
3. Select instruments and procedures for straightness, flatness, roundness, cylindricity, and runout.
4. Interpret the principal parameters of surface texture and select appropriate measuring conditions.
5. Inspect external and internal threads, gears, tapers, and related machine elements.
6. Evaluate measurement results against a drawing specification while considering uncertainty, filtering, datums, and environmental effects.
7. Identify practical, technological, and quality-system limitations of form and surface measurements.

1. Geometric form and its reference framework

1.1 Nominal geometry and error categories

A manufactured feature is compared with an ideal geometric element: a straight line, plane, circle, cylinder, cone, thread, or gear surface. The difference between the actual feature and the ideal feature is the geometric deviation.

The principal categories are:

• Size: numerical magnitude, such as diameter or length.
• Form: deviation of an individual feature from its ideal form, independent of orientation and location. Examples are straightness, flatness, roundness, and cylindricity.
• Orientation: angular relationship to a datum, including parallelism, perpendicularity, and angularity.
• Location: position relative to datums, including position and concentric relationships.
• Runout: variation observed while a feature is rotated about a datum axis; it combines relevant form, orientation, and location effects.
• Surface texture: small-scale deviations, conventionally divided into roughness, waviness, and lay.

Form tolerances normally do not require a datum. Orientation, location, and many runout tolerances do require a datum because the result depends on how the component is established in the measuring system.

1.2 Reference elements and datum simulation

A form error is evaluated by fitting an ideal element to measured data. Common fits include:

• Least-squares element: minimizes the sum of squared deviations. It is statistically useful and stable but is not necessarily the tightest enclosing tolerance zone.
• Minimum-zone element: produces the smallest zone containing all measured points. For a line or plane, the form error is the minimum separation between two parallel boundaries containing the data. For a circle or cylinder, it is the difference between the radii of two concentric circles or coaxial cylinders that contain the data.
• Inscribed or circumscribed element: selected to contain or be contained by the actual feature, useful for functional mating conditions.

The fitting method must be stated because different methods can yield different numerical errors. A coordinate-measuring machine (CMM) software result is not meaningful unless the operator knows the sampling strategy, filtering, fitting algorithm, and datum construction.

A datum is not merely a line or surface drawn on a print. It is established physically by a datum simulator, such as a precision table, spindle, vee-block, or gauge element. Clamping force, contact distribution, cleanliness, and deformation can change the simulated datum and therefore the reported result.

2. Straightness, flatness, roundness, and cylindricity

2.1 Straightness

Straightness describes the departure of a line element from an ideal straight line. It may refer to a surface line in a specified direction or to the derived median line of a cylindrical feature. A straightness tolerance zone consists of two parallel lines separated by the specified tolerance in a two-dimensional case. For an axis or median line, the zone is commonly cylindrical in three dimensions, depending on the specification system.

Typical instruments include a precision straightedge, dial indicator, electronic level, autocollimator, laser interferometer, optical system, and CMM. A dial indicator traversing a surface measures relative displacement, not straightness directly; errors in the reference table, carriage, alignment, and reversal must be controlled.

For a two-dimensional measured profile yᵢ at positions xᵢ, a least-squares straight line can be expressed as:

where y is the fitted ordinate, x is position, a is the intercept, and b is slope. The residual at point i is rᵢ = yᵢ − (a + bxᵢ). A least-squares estimate minimizes ∑rᵢ². This is a fitting model, not automatically the ISO or ASME tolerance evaluation. A minimum-zone evaluation instead minimizes the separation between two parallel lines enclosing all points.

2.2 Flatness

Flatness is the form error of a surface relative to an ideal plane. The tolerance zone is the space between two parallel planes. No external datum is required for flatness, although the workholding arrangement must not distort the part.

A surface plate and indicator provide a practical comparative method. The part is placed on a stable support, a grid is established, and indicator readings are recorded. The result is affected by support points, thermal gradients, dirt, burrs, and indicator alignment. For high accuracy, an autocollimator, electronic level, optical flat, interferometer, or CMM may be used.

If the measured surface points are zᵢ and a reference plane is fitted, the peak-to-valley value is:

where F is the observed zone width in length units. This expression is valid only after the specified reference plane and filtering or sampling procedure have been established. Removing overall tilt is appropriate for a datum-free form evaluation, but not if the drawing intentionally defines a datum-related orientation requirement.

2.3 Roundness

Roundness, or circularity, is the deviation of a cross-sectional line from a true circle. The tolerance zone is bounded by two concentric circles. It must be distinguished from concentricity, which is a location relationship between features.

A roundness tester rotates the workpiece or probe about a highly accurate spindle and records radial variation. CMMs, form tracers, and high-quality rotary tables can also measure roundness. The work must be supported without bending, and the spindle error must be separated or calibrated.

For a measured radial profile r(θ), a circular reference is fitted. Under a minimum-zone method, the roundness error is:

where R_outer and R_inner are the radii of the smallest pair of concentric circles containing the profile. A least-squares circle produces a different value in some cases. Lobing may be identified by plotting radial error against angular position or by harmonic analysis, but filtering and probe bandwidth can suppress or exaggerate lobes.

2.4 Cylindricity

Cylindricity controls the entire cylindrical surface. Its tolerance zone lies between two coaxial cylinders. It combines the effects of circularity in successive sections and longitudinal straightness, but it is not obtained reliably by measuring only one circle and one generatrix.

A cylindricity measurement normally uses a form-measuring instrument or a CMM with multiple circumferential sections and axial traces. Temperature, rotary-axis error, probe qualification, and alignment are important. A least-squares cylinder may be used for analysis, while a minimum-zone coaxial-cylinder evaluation is required when specified by the governing standard or inspection procedure.

2.5 Worked example: interpreting a form result

A shaft is specified with a roundness tolerance of 0.012 mm. A calibrated roundness instrument reports a minimum-zone outer radius of 25.006 mm and inner radius of 24.998 mm. Therefore:

The measured roundness error is 0.008 mm, which is below 0.012 mm. The feature passes the roundness requirement, provided that calibration, filtering, sampling, temperature, and uncertainty are acceptable. This result says nothing by itself about diameter size, cylindricity, or runout.

3. Runout and rotational accuracy

3.1 Circular and total runout

Circular runout is the variation at a specified cross-section as the part is rotated about a datum axis. With a fixed indicator, it is commonly the difference between the maximum and minimum readings during one revolution:

where TIR is total indicator reading and I is the indicator reading. TIR is a range, not a signed displacement and not necessarily twice the eccentricity unless the error is purely circular eccentricity.

Total runout evaluates the entire surface while the part rotates about the datum axis. It can include axial and radial variation over the indicated surface, depending on the feature and setup. Runout is therefore a functional control that may combine form, orientation, and location errors.

A runout result depends strongly on the datum axis. A shaft supported between centers, in a chuck, or on a CMM datum alignment may produce different results. The operator must inspect the datum features first and ensure that contact, seating, and rotational freedom are correct.

3.2 Sources of runout error

Important sources include eccentricity between the functional surface and datum axis, bent shafts, lobing, spindle error, chuck jaw variation, dirt, axial end-play, probe hysteresis, and thermal drift. A reversal test, master artifact, or calibrated spindle can help separate instrument error from workpiece error.

Runout is especially important in rotating assemblies. Excessive runout can cause vibration, uneven bearing loading, seal wear, gear mesh variation, and fluctuating cutting forces. However, reducing runout alone does not guarantee good surface form or balance; these are related but distinct properties.

4. Surface texture metrology

4.1 Texture terminology

Surface texture is the measured topographical character of a surface. It is influenced by tool geometry, feed, cutting speed, vibration, material, abrasives, built-up edge, heat treatment, and finishing processes.

• Roughness consists of relatively closely spaced irregularities.
• Waviness consists of more widely spaced deviations, often caused by vibration, machine deflection, or workpiece distortion.
• Lay is the predominant direction of surface pattern.
• Flaws are isolated unintended imperfections such as cracks, pores, scratches, or pits.

A roughness symbol or numerical requirement on a drawing must be interpreted with its parameter, cutoff or nesting condition, evaluation length, measurement direction, and applicable standard. A value such as Ra 0.8 μm is incomplete if the measurement procedure is not controlled.

4.2 Common parameters

For a filtered profile z(x) about the mean line, the arithmetic mean roughness is:

where Rₐ is in micrometres or another length unit, z(x) is the signed profile deviation from the mean line, and L is the evaluation length. In discrete data:

where n is the number of equally weighted sampled points. The integral and discrete forms assume that the profile has been properly filtered and referenced.

The root-mean-square roughness is:

Rₓ or Rz-type parameters describe peak-to-valley or averaged peak-to-valley characteristics, but definitions vary by standard and edition. Rₐ is not a complete description of texture: two surfaces can have the same Rₐ and very different peak shapes, spacing, bearing properties, or functional performance.

4.3 Contact and optical instruments

A stylus instrument traverses the surface with a diamond tip and converts vertical motion into a profile. It is versatile but can scratch very soft surfaces, cannot enter extremely narrow valleys, and is sensitive to stylus radius, force, speed, cutoff, and traverse direction.

Optical methods include interferometry, confocal microscopy, focus variation, and structured-light techniques. They are non-contact and can provide areal information, but reflectivity, slope, transparency, calibration, data processing, and optical resolution affect the result.

The cutoff or nesting length separates roughness-scale content from waviness-scale content. It must be selected according to the applicable standard and expected texture. The traverse should normally be perpendicular to the lay when assessing machining roughness, unless the specification requires another direction. Measurements should avoid edges, burrs, tool marks not representative of the functional area, and contaminated regions.

4.4 Surface texture and function

Sealing surfaces require a controlled combination of roughness, waviness, lay, and defect population. Bearing surfaces require suitable valleys for lubricant retention without excessive peaks. Fatigue performance is affected by notch-like peaks and directional marks. A lower Rₐ does not always improve function: polishing can remove useful lubricant pockets, and a surface with low average roughness can still contain damaging isolated scratches.

5. Thread metrology

5.1 Thread elements and parameters

Important thread terms include nominal diameter, major diameter, minor diameter, pitch, lead, flank angle, crest, root, pitch line, pitch diameter, and thread form. For a single-start thread, lead equals pitch; for a multi-start thread, lead is the axial distance advanced in one revolution and is greater than pitch.

The pitch diameter is particularly important because it governs flank contact and assembly. A thread can have acceptable major diameter but unacceptable pitch diameter or flank angle. Functional thread gauges combine several effects and are therefore valuable for acceptance, while analytical instruments identify the source of error.

5.2 Inspection methods

External threads may be inspected using:

• major-diameter micrometers or calibrated plain micrometers;
• three-wire methods for pitch diameter;
• optical comparators and profile projectors;
• thread measuring machines or CMMs;
• GO and NO-GO ring gauges.

Internal threads may be inspected using:

• GO and NO-GO plug gauges;
• thread depth or optical systems;
• CMMs with suitable probes;
• specialized internal-thread measuring equipment.

A GO gauge should enter fully or to the specified depth using the prescribed hand force. A NO-GO gauge should not enter beyond the specified limit. Gauge practice must follow the relevant standard; forcing a gauge can damage both gauge and component.

In the three-wire method, three wires of known diameter are placed in the thread grooves and the measurement over wires is taken. The pitch diameter is calculated using a standard-specific relationship involving the measurement over wires, wire diameter, pitch, and flank-angle geometry. Because the exact expression depends on thread form, angle, truncation, and wire selection, the applicable thread standard or calibrated software must be used rather than an assumed universal formula.

5.3 Thread errors and causes

Common errors include incorrect pitch, lead variation, flank-angle error, taper, drunken thread, incomplete form, burrs, and excessive crest truncation. Causes include incorrect tool setting, worn taps or dies, machine backlash, thermal expansion, poor alignment, inadequate lubrication, and chip packing. Functional gauges detect assembly acceptability but should be supplemented by analytical checks during process diagnosis.

6. Gear metrology

6.1 Gear geometry

Gear performance depends on tooth thickness, base-circle geometry, profile, helix, spacing, runout, and surface finish. For an involute spur gear, the circular pitch is:

where p is circular pitch and m is module, both in consistent length units. For a gear with pitch diameter d and number of teeth z:

These relations apply to standard metric involute gear definitions; profile shifts, special systems, and nonstandard gearing require the appropriate design standard.

The base-circle diameter is:

where d_b is base-circle diameter, d is pitch diameter, and φ is the reference pressure angle. The relation assumes an involute gear and a pressure angle measured in the specified reference system.

6.2 Gear inspection

Gear inspection may use master gears, span measurement over a specified number of teeth, tooth-thickness measurement, profile and lead testers, rolling testers, optical systems, and CMMs. The choice depends on whether the purpose is functional acceptance, process diagnosis, or certification.

Important deviations include profile deviation, helix or lead deviation, pitch variation, tooth-to-tooth spacing error, cumulative pitch error, radial runout, and tooth thickness error. Gear standards assign accuracy grades and define evaluation procedures. Inspection must specify the datum axis, measurement diameter, filtering, tooth sampling, and temperature.

A rolling test is highly functional because it examines conjugate action with a master gear, but it may not reveal which individual geometric parameter caused failure. Analytical inspection provides diagnosis but may not reproduce actual mesh behavior. Both approaches are complementary.

7. Tapers, fits, and other machine elements

7.1 Tapers

A taper is a gradual change in diameter along an axis. Taper per unit length is:

where D₁ and D₂ are diameters at two specified axial locations and L is the axial separation. The included angle α of a symmetrical conical surface is related by:

These equations assume coaxial diameter measurements and a straight conical surface. Measurement errors in diameter, location, alignment, and burr condition directly affect the result.

Tapers may be inspected with taper gauges, sine bars, rollers and micrometers, precision CMMs, optical methods, or functional mating tests. A functional test alone may not identify whether the error is angle, size, straightness, or surface damage.

7.2 Cylindrical fits, keyways, and splines

Fits are determined by the relationship between mating sizes and their tolerance zones. Form errors and surface texture can prevent assembly even when two-point size measurements pass. Keyways require control of width, depth, position, parallelism, and burrs. Splines require inspection of tooth thickness, space width, pitch, form, lead, and runout; functional gauges and dedicated analyzers are commonly used.

8. Measurement planning, uncertainty, and data integrity

A sound inspection plan answers five questions: What characteristic is specified? What datum and reference method apply? What instrument has adequate resolution and accuracy? What sampling and filtering are required? How will conformity be decided?

The instrument should have a suitable measurement capability relative to the tolerance. Resolution alone is not accuracy. Calibration establishes traceability to national or international standards, while verification checks performance in the actual application. Temperature is commonly referenced to 20 °C, but the applicable specification governs. Thermal expansion of a feature can be approximated by:

where ΔL is dimensional change, α is the coefficient of linear thermal expansion in K⁻¹, L₀ is the reference length, and ΔT is temperature change in kelvins or degrees Celsius. The approximation assumes uniform temperature and small changes. Workpiece and instrument temperatures should be allowed to stabilize.

For independent standard uncertainty components u₁, u₂, …, uₙ, a basic combined standard uncertainty is:

This root-sum-square form assumes the components are uncorrelated and expressed in compatible units. Correlated components require covariance terms. An expanded uncertainty is often written U = ku_c, where k is a coverage factor selected according to the reporting convention and statistical assumptions.

When a result is close to a tolerance limit, the decision rule must be stated. A conservative guard band may prevent acceptance of a result whose uncertainty interval overlaps the specification boundary. ISO/IEC 17025 principles emphasize competent personnel, calibrated equipment, documented methods, traceability, impartiality, and technically justified statements of conformity.

9. Applied case: runout and surface texture of a pump shaft

Background

The following is an illustrative industrial case. A maintenance manufacturer produces a stainless-steel pump shaft with a bearing diameter, a mechanical-seal diameter, and a threaded coupling end. The drawing specifies diameter size, circular runout relative to the bearing datum, a surface-texture requirement on the seal diameter, and an external thread requirement.

Problem

Several shafts pass diameter inspection but produce seal leakage and vibration during assembly. Operators report that the thread gauges pass and that the seal diameter has an apparently polished appearance.

Analysis

The quality engineer first separates the requirements. Diameter size is checked with a calibrated micrometer. Circular runout is measured with the shaft supported on the specified datum centers, not merely in a production chuck. Surface texture is measured with a calibrated stylus perpendicular to the lay, using the specified cutoff and evaluation length. The thread is checked with GO/NO-GO gauges and then analytically examined if necessary.

The runout readings over one revolution are 0.000, 0.018, 0.006, and 0.011 mm at a seal section. The complete trace gives a maximum of 0.021 mm and a minimum of −0.002 mm. Therefore:

The drawing limit is 0.015 mm, so the feature fails runout even though its diameter is within size limits. A second test shows that chuck mounting produces 0.010 mm while datum-center mounting produces 0.023 mm. The chuck had been masking the datum-related error.

Surface measurements reveal Rₐ = 0.42 μm, below the numerical limit of 0.8 μm, but also reveal a deep longitudinal scratch and excessive waviness near the seal land. The average roughness alone had concealed the functional defect. Thread gauges pass, so thread assembly is not the cause of the leakage.

Solution and outcome

The process team corrects center-hole alignment, verifies the turning-machine tailstock, and adds a datum-center runout check. The seal land is protected during handling and inspected for flaws and waviness, not only Rₐ. A control plan records instrument identification, temperature, setup, operator, filter, direction, and decision rule.

Subsequent shafts meet runout and surface requirements, and leakage is eliminated. The case demonstrates that size, average roughness, runout, and functional performance are related but not interchangeable.

Learning points

1. The measurement setup must reproduce the drawing datum system.
2. A passing size measurement cannot prove form or runout conformity.
3. A single surface parameter cannot describe every functional surface property.
4. Functional and analytical inspections should support one another.
5. Measurement records must be sufficiently complete for repeatability and traceability.

10. Best practices, limitations, and contemporary developments

Best practice begins with drawing interpretation and continues through datum simulation, environmental control, calibration, appropriate sampling, and documented decision rules. Operators should clean parts and contacts, remove burrs without altering the feature, avoid excessive clamping, qualify probes, verify software settings, and repeat suspicious measurements using an independent method.

Limitations arise from inaccessible surfaces, flexible or thin-walled parts, stylus geometry, probe force, sparse CMM sampling, filtering choices, spindle errors, and disagreement between functional and analytical methods. A dense point cloud does not automatically produce a better result; poorly chosen datums or algorithms can create false confidence.

Modern systems combine CMMs, scanning probes, rotary tables, optical sensors, and digital-twin databases. Automated feature recognition and machine learning can identify process trends, classify texture, and predict tool wear. Such systems remain dependent on traceable calibration, representative training data, controlled algorithms, cybersecurity, and human review. AI-generated classifications should not replace a defined standard, validated measurement method, or authorized conformity decision.

Chapter Summary

Form metrology evaluates deviation from ideal lines, planes, circles, and cylinders. Straightness, flatness, roundness, and cylindricity are form properties, while runout depends on rotation about a datum axis and may combine form, location, and orientation effects. Minimum-zone, least-squares, inscribed, and circumscribed fits are not interchangeable, so the evaluation method must be documented.

Surface texture measurement requires control of roughness, waviness, lay, cutoff, evaluation length, probe or optical method, and measurement direction. Rₐ is useful but incomplete. Threads require attention to pitch diameter, pitch, lead, flank angle, and functional gauging. Gear inspection includes profile, lead, pitch, runout, and tooth thickness. Tapers are evaluated from diameter change over axial length and, when needed, by included angle or functional mating.

Reliable inspection depends on datums, calibrated instruments, temperature control, appropriate sampling, uncertainty evaluation, and a clear decision rule. The practical objective is not merely to obtain a number but to establish whether the measured feature will perform its intended function and whether the result is technically defensible.

Key Terms

Form: Deviation of a feature from its ideal geometric form.

Straightness: Deviation of a line element or derived axis from an ideal straight line.

Flatness: Deviation of a surface from an ideal plane.

Roundness: Deviation of a circular cross-section from a true circle.

Cylindricity: Deviation of an entire cylindrical surface from a true cylinder.

Runout: Variation observed while a feature rotates about a datum axis.

Circular runout: Runout evaluated at one specified cross-section.

Total runout: Runout evaluated over the complete specified surface.

Datum: Reference established from a designated feature for measurement or tolerancing.

Minimum-zone method: Fit producing the smallest tolerance zone containing measured data.

Least-squares fit: Fit minimizing the sum of squared deviations.

Surface texture: Topographical character including roughness, waviness, lay, and flaws.

Rₐ: Arithmetic mean roughness of a filtered profile.

Pitch diameter: Effective diameter governing thread flank engagement.

Lead: Axial advance of a thread in one revolution.

Module: Gear pitch diameter divided by number of teeth.

Taper: Change in diameter along an axial length.

TIR: Difference between maximum and minimum indicator readings during rotation.

Measurement uncertainty: Parameter characterizing dispersion of values reasonably attributable to the measurand.

Traceability: Property linking a measurement result to a reference through an unbroken calibration chain.

Review Questions

Short Answer Questions

1. Distinguish form error from size error and give one example of each.
2. Why must the datum system be reproduced when measuring runout?
3. Define roughness, waviness, and lay.
4. What is the functional purpose of a GO and a NO-GO thread gauge?
5. Why is Rₐ alone insufficient to characterize a sealing surface?

Long Answer Questions

1. Explain the principles, instruments, procedures, and limitations of roundness and cylindricity measurement.
2. Compare least-squares and minimum-zone evaluations and discuss why the choice affects conformity decisions.
3. Describe a complete surface-texture measurement procedure for a machined bearing surface.
4. Explain the principal parameters and methods used in gear inspection.
5. Discuss how calibration, temperature, datum simulation, sampling, and uncertainty influence form-measurement results.

Analytical / Application Questions

1. A measured rotating feature has a maximum indicator reading of 0.034 mm and a minimum reading of −0.006 mm. Calculate its TIR and explain what additional information is required before declaring conformity.
2. A taper has diameters 30.000 mm and 29.600 mm measured 100 mm apart. Calculate taper per unit length and the included angle, assuming a symmetrical straight cone.
3. A shaft passes diameter and Rₐ requirements but fails a seal test. Design an investigation that separates size, runout, waviness, lay, and isolated surface-flaw effects.

Multiple Choice Questions with Answers

Question 1

Which characteristic is normally evaluated without requiring an external datum?

A. Flatness

B. Parallelism

C. Position

D. Circular runout

Correct Answer: A. Flatness

Question 2

A rotating feature has a maximum indicator reading of 0.021 mm and a minimum reading of 0.004 mm. What is its TIR?

A. 0.004 mm

B. 0.017 mm

C. 0.021 mm

D. 0.025 mm

Correct Answer: B. 0.017 mm

Question 3

Which parameter is most directly associated with thread flank engagement and assembly?

A. Major diameter only

B. Pitch diameter

C. Surface lay only

D. Overall shaft length

Correct Answer: B. Pitch diameter

Question 4

For a single-start thread, the lead is:

A. Half the pitch

B. Equal to the pitch

C. Twice the pitch

D. Independent of the pitch

Correct Answer: B. Equal to the pitch

Question 5

What does a stylus surface instrument primarily measure?

A. A traced surface profile

B. Thread torque

C. Gear center distance

D. Bulk material hardness

Correct Answer: A. A traced surface profile

Question 6

If a gear has pitch diameter 120 mm and 40 teeth, what is its module?

A. 0.33 mm

B. 3 mm

C. 40 mm

D. 160 mm

Correct Answer: B. 3 mm

Question 7

Which condition is most likely to make a runout result dependent on the setup?

A. Changing the datum axis used for rotation

B. Using consistent temperature

C. Cleaning the contact surfaces

D. Calibrating the indicator

Correct Answer: A. Changing the datum axis used for rotation

Question 8

For a taper with diameters D₁ and D₂ separated by L, taper per unit length is calculated as:

A. (D₁ + D₂)L

B. (D₁ − D₂)/L

C. L/(D₁ − D₂)

D. D₁D₂/L

Correct Answer: B. (D₁ − D₂)/L

Question 9

Which statement about Rₐ is correct?

A. It completely describes all functional surface behavior

B. It is the arithmetic mean of absolute filtered profile deviations

C. It measures only waviness

D. It is always equal to Rq

Correct Answer: B. It is the arithmetic mean of absolute filtered profile deviations

Question 10

A minimum-zone roundness result is based on:

A. The largest diameter measured at two points

B. The smallest pair of concentric circles containing the profile

C. A nominal diameter from the drawing only

D. The average of maximum and minimum readings

Correct Answer: B. The smallest pair of concentric circles containing the profile