Stylus wear and lifetime

How often, or after how many hours play, should I change my stylus? At what point does wear of the stylus affect playback quality? At what point does wear to the stylus start to damage records?

These questions are asked by all enthusiasts of grooved media. Amazingly, for a medium which has been in existence for over 100 years, the answers remain unclear.¹

We have gathered together the limited empirical data which exists (or, at least which we have been able to find) and tried to synthesise some useful advice for the contemporary collector or vinylista.

Happily for the user of Stereo Lab, there exists a novel, modern technique which is reliable and which avoids us having to peer down stylus-inspection microscopes.

Wear and Flats

Friction between the stylus tip and the record gradually wears away both the stylus point and the groove wall as the pressure which exists between them is very great and airborne dust (which is mostly silica) and shed diamond material from the stylus act as an effective abrasive powder between the two surfaces.

At first, the stylus tip touches the groove walls at only two microscopically small points (at least in the case of a spherical or elliptical stylus¹⁴). The friction and resulting wear is concentrated at these points and causes the gradual wearing away of the stylus material creating what are called "flats". The degree of wear of a stylus is expressed in terms of the diameter of these flats.

The figure right is an illustration (from Reference 4) of a conical stylus (top) and elliptical stylus (lower) with and without flats.

The amount of record damage, distortion, and the reduction in frequency-response due to worn styli increases in direct proportion to the size of the flats on the stylus tip. As the dimension of the flat increases, the high-frequency output falls away, and distortion increases (see Appendix 1). But the greater consequence is in increased wear to the record groove scanned by a worn stylus, for the corners formed on the tip by the flats make it an excellent cutting tool as illustrated at (c) left.

First, a misconception

Before we go any futher, we need to address a common misconception. Often, one reads of enthusiasts checking their stylus with a loupe. This is impossible: a loupe or hand-held microscope is OK to check the stylus is clean. But the scale of abrasion is minute, as the scale diagram (right) illustrates. Remember, it is only the very tip of the styus which engages with the groove and the flats are small patches on the very tip.

The data

But this report dates from 1938 and compared steel needles playing 78 RPM records! Clearly, this is useless to the modern enthusiast. We require something more up to date.

Oddly, the BBC does not seem to have repeated these types of test into the LP era. As a broadcast organisation playing hundreds of records daily, the BBC must have developed protocols as to when to change styli right up to the end of the grooved media era in the early 1980s. How did they decide how often to renew styli in order to protect a world-class record library?

The American company Shure, who may usually be relied upon to have researched and published the science behind multiple aspects of phono technology, seems not to have published detailed data from stylus-life testing; although, they clearly did a lot of experimentation and the limited information they did publish remains useful².

The Wear and Care of Records and Styli by Harold D. Weiler³ appears to be the only systematic study of wear to modern, microgroove styli. Backed up by extensive surveying of thousands of users, this booklet collates the results of hundreds of microscopic measurements performed on used styli. It is unique and ought to be more useful than it is. Sadly, the study is dated (1954), pre-stereo and the stylus wear is measured with tracking weights of 7 grams which limits its application to present practice. We have attempted to extrapolate data from Weiler's results with some success (see Appendix 3).

Being wary of wear

".....an injunction to 'change a stylus at a given number of hours' is misleading."

Why are manufacturers and other experts so unwilling to give a clear indication of the useful lifetime of a stylus? Shure make the following points about stylus wear and the difficulties in offering concrete advice.

We have conducted many tests to help us estimate tip life and evaluate the effect of various parameters on tip wear...The size of the tip, trackability or mechanical impedance of the stylus, the record material and recorded content, tracking force, skating compensation, the number of records to be played, the manner in which the records and tips are cleaned, and how frequently they are cleaned are only some of the parameters that must be considered when doing wear tests. The criteria for evaluating wear must also be selected. At what point can we say that a tip has reached the end of its life?

And Arnold Schwartz, Director of Disc Research at CBS Laboratories writing in the New York Times⁸ in 1973 said,

To the oft asked question, “Is there a set number of playing hours before a stylus should be replaced?" my answer is no. Tracking force and other playing conditions vary.

Usefully, Shure do go on to explain² what Schwartz's "playing conditions" are,

In spite of these difficulties, the tests we have conducted over the years have yielded some important results. .... low mechanical impedance; i.e., high trackability and low tracking force (below 1½ grams) [are critical] in achieving long record and tip life.

In other words, it is simply impossible to isolate stylus from the entire dynamic moving system of the phono cartridge when it comes to trying to generalise about stylus wear. Stylus wear (expressed as the size of "flat" developed at the point of contact with the record) is a complex function of playing weight, stylus dimensions, record material, and the dynamic mechanical constants of the pickup referred to the stylus tip. Looked at in this manner, it is easier to see why an injunction to "change a stylus at a given number of hours" is misleading.

Sins of omission

So, what is the vinylista to do?" We know that avoiding using a stylus beyond its useful life is important. But, without a simple rubric like, "renew after 1000 hours" or something similar, how may we act responsibly? Weiler discovered³ that, by the time a stylus exhibited enough audible distortion that it provoked the purchase of a new stylus, it had already travelled well beyond the point of acceptable wear (see Appendix 1 for our determination of this). He says,

This survey and the investigations which followed also indicated that, inadvertently, the vast majority of listeners were severely damaging their valuable collections of records through the continued use of worn styli....... Through ignorance or oversight this important matter has been far too often neglected, even by professionals.

Just a SEK

Back in the 1970s, the manufacturer Shure made a brave attempt to address this issue by encouraging its sales-channel to provide microscopic inspection of styli as a service to the end-user. The retailer could thereby advise the customer as to whether, or not, a stylus needed replacement before significant audible distortion fostered a replacement.

To that end, they produced the Shure Model SEK-2 Stylus Evaluation Kit; a high-quality microscope designed to evaluate the condition of phonograph cartridge stylus tips⁴. Other manufacturers provided similarly adapted microscopes.

How often to inspect

Once accepted that visual inspection of styli is necessary, a further question takes shape: how often should we inspect the stylus? Given that the cartridge may need to be removed from the head-shell or tonearm to make the measurement, this is no trivial query. Of very great help here is some sort of guide as to the rate of wear one can expect, so as to schedule measurements intelligently.

One such exists⁹ from an article by Stanley Kelly in Wireless World magazine in 1969 from which the graph (left) was extracted. Kelly does not attribute this graph, so its provenance is not known. And Kelly himself echoes the other reference here in stating that stylus wear is a complex phenomenon. Nevertheless, the graph does give us somewhere to start.

The data are a surprise. For today's vinylista, only the lower family of curves counts. Nevertheless, it is shocking to realise that a flat with a diameter of 0.00025" (6µm), which is normally considered to be the acceptable stylus-wear limit⁹ is reached for a conical stylus of 0.0007" (18µm) radius in under ten hours if tracked at 5 grams. (This ought to be of some interest to DJs who regularly track records at with conical styli at about 5 grams.)

Even the data at 2 grams indicate that we can only expect a diamond stylus with a scanning radius of 0.0005" (12µm) - a mildly elliptical type - to offer about 500 hours of life. The 0.00025" (6µm) limit (blue line) has been added to Kelly's original graph. To see a large version of this graph, click here.

Time-lapse test

"For those who really can't face the chore of microscopic stylus-wear measurement, Stereo Lab offers a new technique....."

For the user of Stereo Lab who is principally interested in making needle-drop recordings and who wishes to protect records from damage due to a worn stylus, an entirely different approach is available. One which does not require the cartridge to be removed from the head-shell or tonearm, and which eschews the need to squint down a stylus inspection microscope. It is called the time-lapse test.

Appendix 1 derives predictions of the performance fall-off due to a stylus which has reached the end of its working life; a point beyond which it should not be used. Whilst the performance changes aren't great, and certainly difficult to identify as they gradually develop during the lifetime of the stylus, they are at a perceptable level once exposed in an A/B test. The precision signal processing in Stereo Lab allows the comparison to be made by isolating the differences due to stylus wear from those due equalisation and loudness differences.

The time-lapse test simply involves making a reference needle-drop when the cartridge is new¹³ and repeating the process towards the end of the stylus' lifetime (as judged from the graph due to Kelly) to make a direct comparison of the cumulative performance changes. This is accomplished by loading the reference needle-drop recording and the test recording into a multitrack DAW program* and using the solo buttons to "flip" between the two recordings so that a direct A/B comparison may be made.

* Audacity, a free, easy-to-use, multi-track audio editor and recorder which runs on the Mac platform (and, in fact, on Windows and GNU/Linux too) is very suitable for this. Audacity is free software, developed by a group of volunteers and distributed under the GNU General Public License (GPL).

There are a few precautions to making the reference and test recordings and in making the comparison.

• Choose the innermost track on a record for the reference and test track. Ideally select a "pure" musical style involving piano or flute of female vocals and choose a track which is well recorded. We are not testing for high-level distortion, so a ballad, or a slow-movement is best.

• Try to avoid using a record which is well played or which may be played many times during the period between which the reference and test recordings will be made: we want to try to keep the conditions as close as possible for the two tests. The reference recording should be make after a few hours use of the cartridge. Let the stylus "run-in " for about 10 hours before making the reference recording¹³.

• It should go without saying that as little as possible in ancillary equipment ought to change between the two tests. If the computer sound-card/ interface has been changed, or the phono preamp, the before/after comparison may be useless.

• Once the two recordings are available, they should be loaded into the DAW and nudged on the timeline so that they are reasonably synchronised. Perfect synchronisation is not necessary (and may not be possible anyway.) Provided the recordings are within a musical beat or two of one another, that's sufficient.

• It is absolutely imperative that the levels be matched between the two recordings; level trumps all in the human hearing system and you will not be able to make an informed comparison unless the levels are matched to better than 1 dB. Use the loudness processing in Stereo Lab to get this right for you: it will do a much better job than you can manually.

• As to the rest of the processing in Stereo Lab, it should be disabled - with the exception of RIAA equalisation. Neither the reference recording, nor the test recording should have noise-reduction or distortion reduction applied. Automatic azimuth/balance ought not to be checked in the Phono Preferences tab..

• Appendix 1 covers the changes you need to be listening out for and which are the signs of a stylus nearing its end-of-life. Once the stylus is worn, there will be a fall in treble energy with a "blunting" of the high harmonics. If the reference recording sounds noticeably "brighter" suspicion should fall immediately on the stylus. Similarly, if you notice a hint of a "fizzing" or "grumbling" type of distortion at low level, that is a sign of the distortion we expect from a worn tip. Sometimes this manifests itself as a general "dirtiness" to the audio as compared with the reference. If the two effects, the HF loss and "dirty" distortion are combined, it's a very clear sign that it's time to change stylus.

• If you use a stylus with an advanced profile (Shibata, Obata, Quadrahedral, Pramanik or line-contact types), be sure to read Appendix 2.

Frequency response

In order for us to appreciate the effect of "flats" upon sound production (firstly we consider frequency-response), we can use the very simple simple mathematical model of a boxcar filter. The idea being that the stylus flat acts like a moving-average filter as it scans the disc modulation.

Into this simple model, we can substitute some dimensions such as we encounter with real styli flats and grooves. We'll call the diameter dimension (in micrometres) of the flat f and give it the dimension of 6µm, which is normally considered to be the wear limit of a 18µm conical (spherical) stylus. The innermost groove of an LP (where the flats will cause the greatest high-frequency loss), the velocity of the stylus in the groove is ≈ 0.2m/s. And thus the total delay (D) from the leading-edge to the trailing-edge is,

   D = f ÷ 0.2 = 30µS.

This is equivalent to about 3 sampling periods at 96kHz. So we can formalise this by imagining three versions of the sampled audio, separated by one sample period and mixed together (as shown in the diagram of the boxcar filter).

The transfer-funtion |H(ω)| of a boxcar flter of this type has the form of |^sin(x)⁄_x|.

Applying the appropriate scaling, we can plot the graphs below, the left-hand graph showing the wideband response (and the classic |^sin(x)⁄_x| shape). And the right-hand graph illustrating the effect in the audio band alone.

The effect is small, about 1dB at 8kHz, 3dB down at 14kHz. When the audio pre and post this "apperture" filter are A/B compared, careful listening will reveal this degree of difference. However, it would be impossible to perceive this change developing slowly over time. Weiler is right that it is easy to prolong use of a stylus beyond the danger point that it is causing harm to the record without access to an A/B comparison as we have in the time-lapse test.

Distortion

Properly to do this justice would be a complicated mathematical problem. However, we can get an idea of the magnitude of the problem by considering the degree of error which a flat (of 6µm diameter, which is normally considered to be the acceptable stylus-wear limit⁹) introduces. Simplifying this even more, rather than consider the abraded stylus material as a three dimensional spherical cap, we consider the eroded matter as a circular segment.

A circular segment (in green) is enclosed between a secant (the dashed line of dimension c) and the arc whose endpoints equal the chord's. Let this (green) segment be the amount of abraded stylus.

The sagitta (dimension h) in the diagram is related to the other dimensions by,

Thus, for a stylus with a radius of 18µm and a 6µm flat,

    h = 0.25µm

If we imagine this worn stylus tracking a gradual groove modulation, the stylus movement will "pause" as the flat touches the peaks of the curvature instead of following them. (Think of a flat-faced cam-follower where this pause in the movement is called dwell.) This condition will apply to all frequencies, but the dwell will become a greater proportion of the cycle as recorded wavelength falls, which is to say; at higher frequencies and towards the centre of the record.

Interestingly, and unlike virtually all the other distortion mechanisms in record reproduction, the dwell discontinuity will be independent of recorded amplitude and, because the pickups are velocity sensitive, the discontinuity will not be at signal peaks, but at the zero crossings.

Standard recording level is equivalent to a displacement of 11.2µm peak of a sinewave modulated groove at 1kHz, or 22.4µm pk-pk. The stylus abrasion for a 6µm flat (at its deepest point of abrasion, as we calculated) is 0.25µm. If we refer this to standard recording level for a 1kHz tone we can predict the rough magnitude of the distortion.

    ^0.25⁄_22.4 ≈ 1%

This amount of distortion is detectable on music programme if you know what to listen for and, especially, if you can A/B compare the pre- and post-distorted versions. (This confirms the observations of both Weiler and Schwartz^3,8.) As with frequency-response modifications, the distortion introduced by a worn stylus is below a level at which we can be sure of identifying a component if it "creeps up on us" which might lead us to damaging records (consider the points marked with red arrows in the graphic above).

Keeping count

We use a simple, mechanical thumb-operated counter to keep track of stylus use. We assume that one side of an LP is 20 minutes so that a count of 3 represents one-hour of stylus use.

"..... remember that [the Shibata] profile was not conceived to improve the performance from standard stereo records."

Introduced in the last decade of the dominance of records as the main medium for recorded music, CD-4 pushed the envelope of what was deliverable from a grooved medium and no discussion of stylus life (or indeed of record wear) would be complete without considering the jumbled legacy CD-4 quadraphonic has left.

The team at Pspatial Audio has probably spent more time looking at CD-4 literature than anyone else in the last few years as we developed our CD-4 software decoder. We feel, because of this, we are well placed to look at the legacy of this technology which, we feel, is often misunderstood and is the root of some enduring urban myths.

Stylus-shape and record wear

Tracing the CD-4 subcarrier information from the grooves of a humble record is no straightforward task. Sideband information extends to 45kHz, a frequency about three times that recorded on a normal stereo disc.

To record this modulation, it was necessary to cut the master acetate at one third speed, and to recover it required a stylus which could scan tiny wavelengths (especially on inner grooves). Norio Shibata of JVC designed such a remarkable thing. But, it's important to remember that this stylus profile was not conceived to improve the performance from standard stereo records. Neither were the copies and variations of the Shibata known as the: Obata, Quadrahedral or Pramanik; all of which possess the essential quality of a large vertical bearing radius compared with the scanning radius. None of these styli were designed to ameliorate performance below 15kHz.

Nevertheless, over the intervening years, there's plenty of evidence that there are indeed some gains in frequency-response and distortion from using a Shibata type stylus to playback a stereo record; in line with the considerations predicted by the geometry of a narrower scanning radius. To this degree, the perpetuation of Shibata styli and its derivatives after the failure of CD-4 is reasonable and justified.

The widely-held opinion that a stylus with a larger load bearing radius does less harm to records also appears to come from work on the wear of CD-4 records¹⁰. Here the inference that a Shibata stylus creates less wear on a stereo record is not entirely secure. It is certainly the case that a new stylus with a Shibata profile will wear CD-4 records more slowly than will a conical or elliptical type¹⁰. In fact, it is essential only to play CD-4 records with a Shibata type stylus; as explained on this page. But the original research did not consider the effects upon stereo records. And it did not examine record wear effects once the stylus itself was worn.

Any component with an edge a few multiples of the wavelength of light has to be susceptible to chipping. And there is some evidence of this susceptibility dating from after the collapse of the quadraphonics project, from one of the pioneers of using a scanning electron microscope (SEM) in phono stylus development, who reported,

Many diamond shapes have been produced in the past few years which have shown excellent performance when they were new and their contact area was hardly worn. But, as we photographed [using the SEM] several of the worn exotic shapes of styli [Shibata], we have discovered that their contact areas have changed drastically producing even greater groove deformation than worn ordinary spherical or elliptical tip styli"..... Note the furrows caused by uneven wear in the figure (right) below. ⁷.

Stylus-shape and stylus wear

"..... it's understandable, if you believe records are worn more slowly with a Shibata type stylus, to infer that the stylus must be worn more slowly: but it's an unsubstantiated deduction."

The original CD-4 research¹⁰ did nothing to study the wear of the stylus, the focus was entirely upon record wear. Now, it's understandable, if you believe records are worn more slowly with a Shibata type stylus, to infer that the stylus must be worn more slowly: but it's an unsubstantiated deduction. We know of no published, systematic, empirical study that has proved that claim. On the other hand, we have evidence to the contrary. Shure ran careful, automated large-scale wear tests² to examine comparative wear of styli with different tip shapes and concluded,

Included among our tip wear tests are studies of the relative tip life of long contact tips versus conventional biradial tips. These tests were conducted on commercial changers of the same model, and each stylus played its own record continuously, tracking at 2.0 grams force. Photographs of the contact regions at 600x magnification were used to evaluate the rate of wear. A comparison of styli with long contact tips and styli with biradial tips revealed no significant difference in the rate of wear between the two groups as a whole, although there were differences between individual tips.

Stanton produced this astonishing picture of a worn Shibata stylus at ×3000 magnification⁷. They note that, the delta shape contact area scrapes the bottom of the groove. Look also at the scoring of the edge.

Appendix 3 - Cross referencing Kelly and Weiler's data

It's satisfying to note that, with a bit of mathematical fiddling, Kelly's data⁹ may be compared (moderately favourably) with that of Weiler³. We read from Kelly's graph that the wear at 5 grams is about 40× faster than that due to 2 grams. Now,

    (⁵⁄₂)⁴ ≈ 39

Or in words, wear appears to be related by the fourth-power of downforce.

Extrapolating Weiler's data as we have done in the graph (right), we can see that, according to Weiler, a 0.001" stylus wears to a 0.0005" flat in about 70 hours with a 7 gram vertical tracking weight. And Kelly's data indicates a 0.001" stylus wears to a 0.0005" flat in about 400 hours.

So, if we apply the same downforce correction,

    (⁷⁄₅)⁴ ≈ 4.. , and 4 × 70 ≈ 280 hours

Hardly a perfect match, but near enough to suggest that both data sets (from 15 years apart) are not entirely inconsistent.

Notes and References

1. The Finish Line for Your Phonograph Stylus. Bodell, M. https://thevinylpress.com/the-finish-line-for-your-phonograph-stylus/

2. The Stylus Tip and Record Groove - The First Link in the Playback Chain. B. W. Jakobs and S. A. Mastricola, from High Fidelity Phonograph Cartridge - Technical Seminar NYC 1978

3. The Wear and Care of Records and Styli. Harold D. Weiler. Climax Publishing Company NYC 1954

Weiler's stylus images are impressive (example, right). He was able to obtain microphotographs using a microscope with a reticule marked in tenths of thousandths-of-an inch (2.5µm scale) with the stylus viewed from the side. With such a tool, it is possible precisely to determine the wear radiï.

The SEK-2 product is a decent quality 100-200× monocular microscope (made by Swift Instruments) combined with a micro-metrically controlled x- and y-plane horizontal stage and a two-lamp stage illuminator. The SEK-2 is designed so that the cartridge under inspection lays on its back (in its most stable orientation) and the objective lens looks down upon the tip of the stylus.

The procedure for evaluating the stylus for wear using the SEK-2 is the following. Once the stylus has been cleaned and brought into position and focused at the lower magnification (100×), the zoom eye-piece is rotated to allow the stylus to be inspected at 200× magnification.

At this relatively high degree of magnification, the depth of field collapses so that the form of the stylus is not visible at all, but focus may be made on the wear spots (the "flats") which - illuminated by the lights at either side - are visible as bright spots of light which Shure called "cat's eyes". These cat's eyes are the key to judging the degree of wear and Shure provided the figures (left) to illustrate stylus conditions (for a variety of stylus shapes) at this magnification.

As the figures illustrate, it is not the size of the cat's eyes so much as their proximity to one another which betrays the degree of wear suffered by the stylus. We must beware the cross-eyed cat!

Should I get one?

A Shure Model SEK-2 Stylus Evaluation Kit does appear on on-line auction sites from time to time, but they are now rare and Shure no longer manufacture the unit (in fact, Shure exited the phono business entirely in the summer of 2018⁵).

The existence of inexpensive, USB microscopes with a magnification up to 200× means that the construction of a DIY inspection microscope of a quality close to that of the Shure SEK-2 is within the reach of most enthusiasts. We have found that, provided the microscope has secure mount and stage and that two strong lights are provided above and from either side of the microscope stage, it is quite possible to re-create Shure's "cat's eyes" patterns.

Unfortunatley, as Shure's sample pictures of good, moderate and bad styli illustrate, the degree of wear isn't especially unambiguous using this technique. Shure's ingenious idea with the SEK-2 isn't to judge the wear directly, but to examine a secondary effect; the reflection on the flat of a bright light. With the benefit of experience, it is possible get quite good at determining the degree of stylus wear. But for the neophyte, examination can be frustratingly indeterminate using Shure's technique.

5. Shure discontinued all of its phono products from summer 2018. Shure had been making cartridges since the 1940s including the first cartridge capable of playing both 33 and 78 rpm records. At the height of its production, Shure was manufacturing roughly 28,000 cartridges a day, 25,000 of which were made at its phonograph production plant in Arizona.

6. GRAMOPHONE REPRODUCTION: WEAR OF NEEDLES AND COMMERCIAL RECORDS BBC Report Serial No. 1938/7

7. Role of Scanning Electron Beam Microscope in Disc Recording. Alexandrovich G. Presented at the 58th Convention of the AES November 1977.

8. The First Component Of Your Hi-Fi System—Your Phono Stylus. Schwartz, A. New York Times December 2, 1973, Page 708

9. Stereo Gramophone Pickups A review of the various types of transducer available. Kelly, S. Wireless World December 1969

10. Development of a Compound for QuadraDiscs, Bogantz G.A. and Khanna, K. JAES Jan/Feb 1975 (Originally given as a paper at the 49th AES Convention - September 1974)

11. Standard recording level is equivalent to a displacement of 11.2µm peak of a sinewave modulated groove at 1kHz. Velocity can be calculated using the formula, v(t) = 2 . π . f . a, where a is the peak amplitude and f is frequency. Thus,

2 × π × 1000 × (11.2 × 10^-6) = 0.07m/s = 7cm/s peak or 5cm/s RMS

13. One interesting aspect of Kelly's wear graph (and indeed those of Weiler) is the logarithmic time-scale. We believe this is because, initially, the friction takes place between a small area of the stylus (at least for a spherical or elliptical type), the creation of flats happens quickly; within the first few hours. Then, as the flat develops, and more diamond material is exposed to the friction, the wear slows down; at least when judged by the diameter of the flats. (Remember the area of a circle is related to the square of its radius.) Interestingly, the initial period of wear from an almost perfect point to a very small flat would seem to be consistent with the timescales enthusiasts discuss for breaking-in a phono cartridge.

14 . Even in the the case of a spherical or elliptical stylus, it isn't true that the stylus tip touches the groove walls at only two points. Instead, the plastic deforms due to the pressure exerted by the stylus so that two areas of contact develop between the stylus and the groove. (In calculating this, it's safe to assume that the diamond stylus itself is entirely rigid which it is in relation to the relatively soft plastic of the record.)

Towards the end of his tragically short life (and after he had confirmed Maxwell's prediction of the existence of electromagnetic waves) Hertz published the solution to the problem of contact of a sphere with a plane for elastic bodies (H. Hertz, Gesammelte Werke, pp. 155-196, Leipzig, 1895) such that the radius of the contact surface is given by,

    a = [ ⅓ × ^(RW) ⁄ _{(E. sinβ)} ]^⅓

where,

W = stylus force (lb).
R = stylus radius (in.).
β = ½ the groove angle.
E = Young's modulus of elasticity.
  Poisson's ratio here is assumed to be ⅓.
So, with a stylus of radius 0.0007", vinyl with a Young's modulus of 490,000 psi ¹⁵, with a tracking force of 2 grams (or 0.00440925 lbs), the the radius of the contact surface is,

    = 0.00014" = 3.6µm, or a contact area diameter of 7.2µm.

Considering this dimension, it's easy to see why a flat of 6µm diameter on the diamond's surface is considered the reasonable wear limit.

15. Young's modulus figure for vinyl derived from information in a letter from D. A. Barlow in Letters to the Editor JAES March 1971. In the letter, Barlow states he measured 3.34 × 10¹⁰ dyne/cm² as the Young's modulus and ⅓ as Poisson's ratio for contemporary PVC co-polymer plastic records. This is equivalent to 484,426 psi. In earlier work (Groove Deformation in Gramophone Records Wireless World April 1964), Barlow had used the approximate figure of 3.00 × 10¹⁰ dyne/cm².

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