Fabric Geometry: Peirce’s Woven Model, Munden’s Knit Loop and Tubular Braids

This chapter answers the third part of Section XE9.3, Fabric Manufacture, Structure and Properties, of XE9 Textile Engineering and Fibre Science, one of the nine optional sections of the Engineering Sciences (XE) paper: “Peirce’s equations for plain woven fabric geometry; Thickness, cover and maximum set of woven fabrics; Geometry of plain weft knitted loop; Munden’s constants and tightness factor for plain weft knitted fabrics; Geometry of tubular braids.” It states Peirce’s circular-thread model and its equations and solves the two special cases that give closed forms, then derives crimp, thickness, cover, maximum set and fabric mass; it gives the plain weft-knit loop relations of Munden, the tightness factor and the mass per unit area; and it works out the helix geometry of a tubular braid, including braid angle, yarn length, cover and the angle of maximum volume. Every numerical states its constants in the stem; the answers are crimp, thickness, maximum set, areal density of woven and knitted fabrics, and braid angle.

1. Peirce’s model of plain-woven cloth

Geometry turns a fabric specification (yarn diameter, thread spacing) into crimp, thickness and cover. Peirce’s model (1937) idealises plain weave: each thread has a circular cross-section, does not flatten and does not stretch; its axis follows circular arcs and straight lines; and warp and weft threads touch at the crossings. Warp is system 1 and weft is system 2. For each system, d is the diameter, p the spacing between neighbouring threads of that system, l the length of one thread between the planes through the axes of two adjacent threads of the other system, h the crimp height (peak-to-peak displacement of the thread axis normal to the cloth) and θ the maximum inclination of the axis to the cloth plane. Put D = d₁ + d₂.

Peirce’s equations for a plain weave (circular-thread model)
RelationMeaning
h₁ + h₂ = DAt a crossing the two touching threads have axes D/2 apart, so the crimp heights share one fixed total
p₂ = (l₁ − D θ₁) cos θ₁ + D sin θ₁Horizontal extent of thread 1 between two weft threads: a straight part plus two arcs of radius D/2
h₁ = (l₁ − D θ₁) sin θ₁ + D (1 − cos θ₁)Vertical rise of thread 1 over the same stretch
p₁ and h₂ from the same two equations with subscripts 1 and 2 interchangedThe weft obeys the same geometry about the warp
c₁ = l₁/p₂ − 1, c₂ = l₂/p₁ − 1Crimp, here taken as extra thread length over the fabric length

The six unknowns l₁, l₂, h₁, h₂, θ₁, θ₂ are tied by five relations, so with d₁, d₂, p₁, p₂ known one more datum (a crimp, an angle, or a jamming condition) fixes the structure. The solution generally needs a numerical trial on θ, but two cases give closed forms. Warp-crimped, weft straight (h₂ = 0, θ₂ = 0, so h₁ = D): the h₁ equation gives (l₁ − Dθ₁) sin θ₁ = D cos θ₁, hence p₂ = D ÷ sin θ₁ and l₁ = D (θ₁ + cot θ₁), so crimp c₁ = θ₁ sin θ₁ + cos θ₁ − 1. Balanced (equal threads, h₁ = h₂ = D/2, θ₁ = θ₂ = θ): p = D (1 − ½ cos θ) ÷ sin θ and l = D [θ + (cos θ − ½) ÷ sin θ].

Worked example. Straight weft, θ₁ = 30° (0.5236 rad), D = 0.40 mm. Then p₂ = 0.40 ÷ 0.5 = 0.80 mm, l₁ = 0.40 (0.5236 + 1.7321) = 0.902 mm, and crimp = 0.5236 × 0.5 + 0.8660 − 1 = 0.1278, that is 12.8% (check: 0.902 ÷ 0.80 − 1 = 12.8%). For the balanced case at θ = 30° the formulas give p = 1.134 D and crimp 10.7%.

⚠️ Crimp interchange
Because h₁ + h₂ = D, forcing one system to bend more makes the other bend less. If ends per cm are raised with picks per cm fixed, the warp spacing p₁ shrinks, the weft tends to bend more over the closer warp (weft crimp rises) and the warp crimp falls. Check which system a question calls crimp 1 or 2, and whether crimp is based on the fabric length (as here) or on the thread length.

2. Yarn diameter, thickness and cover

Yarn diameter. For a round yarn of linear density tex (mg per m) and yarn density ρ (g/cm³), the cross-section is A = tex ÷ (1000 ρ) mm², so d = √(4A/π) = 0.0357 √(tex/ρ) mm. Spun yarns are porous, so ρ is the yarn’s own packed density, well below the fibre density, and it must be stated in the problem. Doubling the tex raises d by √2.

Thickness. The axis of thread i ranges over h_i, and the thread adds half a diameter above and below, so the thread’s envelope is h_i + d_i; the cloth thickness is the greater: T = max (h₁ + d₁, h₂ + d₂). A balanced plain weave of equal threads (h = d) has T = 2d. With the weft straight, T = D + d₁. Cover. The fractional cover of a system is e = d ÷ p. The two systems overlap, so the cloth cover is e = e₁ + e₂ − e₁ e₂, not the sum. For e₁ = 0.50 and e₂ = 0.40, e = 0.70.

3. Maximum set and fabric mass

Maximum set. Threads of one system cannot be closer than p = d (cover 1). In the circular-thread model a balanced plain weave jams earlier. Differentiating p = D (1 − ½ cos θ) ÷ sin θ gives a minimum where cos θ = ½, θ = 60°, at which the straight length l − Dθ is zero and p_min = (√3 / 2) D. For equal threads D = 2d, so p_min = √3 d = 1.73 d and the cover of each system is at most d ÷ p = 0.577, so the cloth cover is at most 2(0.577) − 0.577² = 0.82. The crimp at this limit is 20.9%. Maximum threads per cm = 10 ÷ p_min (mm). Real fabrics set closer by flattening the threads, which this model excludes.

Worked example (maximum set). Yarn 20 tex with ρ = 0.80 g/cm³: d = 0.0357 √(20 ÷ 0.80) = 0.0357 × 5 = 0.178 mm. Balanced limit p_min = 1.732 × 0.178 = 0.309 mm, so at most 10 ÷ 0.309 = 32.4 threads per cm in each direction. Fabric mass. With e ends and n picks per cm, tex₁ and tex₂, and crimps c₁ and c₂ (on fabric length): mass per m² = 0.1 [e (1 + c₁) tex₁ + n (1 + c₂) tex₂] g.

4. Plain weft-knitted loop, Munden’s constants and tightness factor

A plain weft-knitted loop is a space curve held by the neighbouring loops. Idealised models of it (Peirce’s loop of arcs and straight lengths, and later three-dimensional models) fix the loop length l, the yarn length in one loop, and relate the loop’s height and width to l and the yarn diameter. In practice the dominant variable is the loop length l (also called stitch length), because the fabric dimensions after relaxation depend mainly on l, not on the knitting tension. Let c be courses per cm, w wales per cm and S = c w the stitch density (loops per cm²).

Munden’s relations. In a given relaxed state, Munden found that c l, w l and S l² are constants of the structure: c = K_c ÷ l, w = K_w ÷ l, S = K_s ÷ l², with K_s = K_c K_w. In the dry-relaxed state for common plain cotton fabrics the constants are of the order K_c ≈ 5, K_w ≈ 4 and K_s ≈ 20, but they depend on the state of relaxation, fibre and yarn, so a problem must supply them. The ratio c ÷ w = K_c ÷ K_w is the loop shape factor and it is independent of l. Doubling l halves c and w and cuts S to a quarter.

Tightness factor. TF = √tex ÷ l measures how much of the loop’s area the yarn fills, because yarn diameter is proportional to √tex. Its value depends on the length unit: with l in cm, typical values are near 14 to 15; with l in mm they are ten times smaller (about 1.4 to 1.5), so the unit must be stated. A larger TF means a tighter, denser, stiffer fabric. Mass per unit area. Each loop holds l of yarn, so areal density = S × l × tex: in g/m², with l in cm and S per cm², areal density = 0.1 S l tex = 0.1 K_s tex ÷ l.

Worked example. Yarn 20 tex, l = 3.0 mm = 0.30 cm, K_c = 5.0, K_w = 4.0. Then c = 5.0 ÷ 0.30 = 16.7 per cm, w = 4.0 ÷ 0.30 = 13.3 per cm, S = 16.7 × 13.3 = 222 per cm². TF = √20 ÷ 0.30 = 14.9 (l in cm). Areal density = 0.1 × 222 × 0.30 × 20 = 133 g/m², as 0.1 × 20 × 20 ÷ 0.30 gives. At fixed tex, a larger l makes a lighter, more open fabric.

⚠️ Relaxation state and unit slips
Munden’s constants are not universal: a fabric straight off the machine is distorted by knitting tension, and only after full relaxation do different fabrics of the same fibre and yarn reach near-equal constants. Keep to one length unit throughout (cm for the areal-density formula above), and do not mix the stitch length of one course with the length of yarn per unit area.

5. Geometry of tubular braids: helix, pitch and braid angle

In a biaxial tubular braid each yarn follows a helix on a cylinder of diameter D (measured at the yarn axis). Cut and unroll the cylinder: one turn of a yarn becomes the hypotenuse of a right triangle whose sides are the circumference πD and the pitch h (the axial advance for one full turn). With the braid angle θ measured from the braid axis: tan θ = πD ÷ h, and the yarn length in one turn is L = √[(πD)² + h²] = πD ÷ sin θ = h ÷ cos θ. The yarn is longer than the braid by the factor 1 ÷ cos θ.

From the machine. If a yarn circles the axis at f revolutions per second (a mean rate set by the carrier speed) and the braid is taken up at v, the pitch is h = v ÷ f and tan θ = π D f ÷ v. Raising the take-up speed lowers tan θ, giving a smaller braid angle, a longer pitch and a more axial, stiffer-along-the-axis braid. For example D = 10 mm, f = 0.50 rev/s and v = 20 mm/s give h = 40 mm and tan θ = 10π ÷ 40 = 0.785, θ = 38.1°.

6. Braid cover, constant yarn length and the neutral angle

Cover. Let N be the total number of yarns (carriers), half in each direction, each of width w. Around the circumference, one set’s yarns are w ÷ cos θ wide, so the fractional cover of one set is F = N w ÷ (2π D cos θ), and treating the two sets as independent layers, the braid cover is CF = 2F − F². Cover rises as θ rises, since the yarns lie more circumferentially, and as D falls. Example: N = 16, w = 1.0 mm, D = 10 mm, θ = 40°: F = 16 ÷ (2π × 10 × 0.766) = 0.332 and CF = 0.664 − 0.110 = 0.554.

Constant yarn length and the neutral angle. If the yarn length per turn L is fixed and the braid is stretched or compressed, πD = L sin θ and h = L cos θ. The enclosed volume per turn is ∝ D² h ∝ sin² θ cos θ. Setting its derivative to zero, 2 sin θ cos² θ − sin³ θ = 0, gives tan² θ = 2, so θ = 54.7° maximises volume. Pulling a braid axially lowers θ (it narrows and lengthens); squeezing it along the axis raises θ toward 54.7°, so a pressurised braided hose tends to settle at this angle, the basis of braid-sleeve actuators.

Key takeaways

  • Peirce’s circular-thread model has h₁ + h₂ = D = d₁ + d₂, with p = (l − Dθ) cos θ + D sin θ and h = (l − Dθ) sin θ + D (1 − cos θ) for each system. One more datum fixes the structure. With the weft straight p₂ = D ÷ sin θ₁ and crimp = θ sin θ + cos θ − 1.
  • Crimp interchange: more bend in one system means less in the other. Thickness T = max (h₁ + d₁, h₂ + d₂), which is 2d for a balanced plain weave. Cloth cover = e₁ + e₂ − e₁ e₂ with e = d ÷ p.
  • Yarn diameter d = 0.0357 √(tex ÷ ρ) mm. The balanced circular-thread limit is θ = 60° with p_min = √3 d, so at most 10 ÷ p_min (mm) threads per cm and cloth cover 0.82. Woven mass = 0.1 [e (1 + c₁) tex₁ + n (1 + c₂) tex₂] g/m².
  • Plain weft knit: c = K_c ÷ l, w = K_w ÷ l, S = K_s ÷ l². Doubling l halves c and w, quarters S. Tightness factor = √tex ÷ l, so state the unit of l. Areal density = 0.1 S l tex = 0.1 K_s tex ÷ l (g/m², l in cm).
  • Tubular braid yarn is a helix: tan θ = πD ÷ h, yarn length per turn L = πD ÷ sin θ = h ÷ cos θ. From the machine tan θ = π D f ÷ v, so faster take-up gives a smaller braid angle.
  • Braid cover per set F = N w ÷ (2π D cos θ) and CF = 2F − F², rising with θ. At fixed yarn length the enclosed volume is maximal at θ = 54.7° (tan² θ = 2).

Practice questions (17)

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.

  1. In Peirce’s circular-thread model of a plain weave in which warp and weft touch at every crossing, the sum of the two peak-to-peak crimp heights h₁ + h₂ equals which quantity?

    1. The larger of the two thread diameters, whichever system it belongs to
    2. Half the sum of the two thread diameters, (d₁ + d₂) ÷ 2
    3. The sum of the two thread diameters, d₁ + d₂
    4. The sum of the two thread spacings in the cloth, p₁ + p₂
    Show answer

    Answer: C — The sum of the two thread diameters, d₁ + d₂

    At one crossing the touching threads’ axes are (d₁ + d₂)/2 apart, and at the neighbouring crossing the order is reversed, so the two peak-to-peak heights add up to d₁ + d₂ = D. Half of that is the axis separation at a single crossing, which is the tempting wrong answer. Spacings do not enter this contact condition.

  2. A plain-woven cloth is made with the same weft and the same warp yarn, but the ends per cm are increased while the picks per cm are held constant. In Peirce’s model what is the usual tendency of the crimps?

    1. The weft crimp rises and the warp crimp falls, as the closer warp forces the weft to bend more
    2. Both crimps rise together, because the cloth is now more tightly packed in the warp direction
    3. The warp crimp rises and the weft crimp falls, because the warp now has less room to lie flat
    4. Neither crimp changes at all, because crimp depends only on the yarn diameters
    Show answer

    Answer: A — The weft crimp rises and the warp crimp falls, as the closer warp forces the weft to bend more

    Since h₁ + h₂ = D is fixed, a higher ends density, which crowds the warp and raises the weft’s bending, shifts crimp from the warp to the weft: crimp interchange. Both rising would need the sum of heights to grow, which the contact condition forbids. Crimp depends on the thread spacings too, not only on the diameters.

  3. The warp cover of a cloth is 0.50 and the weft cover is 0.40, each defined as thread diameter ÷ thread spacing. What is the fractional cloth cover when the overlap of the two systems is allowed for as e₁ + e₂ − e₁ e₂?

    1. 0.90
    2. 0.20
    3. 0.45
    4. 0.70
    Show answer

    Answer: D — 0.70

    Cloth cover = 0.50 + 0.40 − 0.50 × 0.40 = 0.70. The plain sum 0.90 counts the overlapped area twice, which is the common error. The product 0.20 is only the overlap, and 0.45 is the mean, which has no meaning here.

  4. The diameter of a round yarn follows d = 0.0357 √(tex ÷ ρ) mm for a fixed yarn density ρ. If the linear density is raised from 15 tex to 60 tex at the same ρ, what happens to the diameter?

    1. It becomes four times larger
    2. It doubles
    3. It becomes sixteen times larger
    4. It rises by 50%
    Show answer

    Answer: B — It doubles

    The diameter varies as √tex, and the tex rises by a factor of 4, so d doubles. Four times larger would apply if diameter were proportional to tex, which is the usual slip.

  5. A plain weft-knitted fabric is made from a given yarn, and the loop length is doubled while the fabric is brought to the same relaxed state. Using Munden’s relations, how does the stitch density per cm² change?

    1. It halves, because the courses per cm halve
    2. It falls to one quarter, because both c and w halve
    3. It is unchanged, because the yarn is the same
    4. It doubles, because each loop now holds more yarn
    Show answer

    Answer: B — It falls to one quarter, because both c and w halve

    S = K_s ÷ l², so doubling l reduces S to a quarter, since both courses per cm and wales per cm halve. Halving alone accounts for only one of the two directions. Having more yarn per loop makes each loop larger, not more numerous.

  6. A tubular braid is made at a constant carrier speed with a braid angle of 45° to the axis. The take-up speed is doubled while the braid diameter stays the same. What is the new braid angle?

    1. 22.5°
    2. 26.6°
    3. 30.0°
    4. 63.4°
    Show answer

    Answer: B — 26.6°

    tan θ = π D f ÷ v, so doubling v halves tan θ from 1 to 0.5, and θ = arctan 0.5 = 26.6°. Halving the angle itself (22.5°) is the tempting slip. 63.4° corresponds to tan θ = 2, the effect of halving the take-up instead.

  7. The helical yarns of a tubular braid have a fixed yarn length per turn, and the braid is deformed so that its angle θ to the axis varies. At which angle is the volume enclosed per turn greatest?

    1. 45°, because the pitch and the circumference are then exactly equal
    2. 60°, because the helix is then closest to the circumferential direction of the tube
    3. 30°, because the yarn is then closest to the axis of the tube
    4. 54.7°, where tan² θ = 2
    Show answer

    Answer: D — 54.7°, where tan² θ = 2

    Volume per turn ∝ D² h ∝ sin² θ cos θ, which is maximum when 2 cos² θ = sin² θ, that is tan² θ = 2, θ = 54.7°. At 45° the pitch equals the circumference, but this does not maximise the product. The extremes 30° and 60° both give smaller volumes than 54.7°.

  8. In Peirce’s model a plain weave has its weft straight (h₂ = 0), and the warp makes a maximum angle θ₁ with the cloth plane. If θ₁ is increased while the diameters are unchanged, what happens to the weft spacing p₂ and the warp crimp?

    1. p₂ decreases as D ÷ sin θ₁ and the warp crimp increases
    2. p₂ increases and the warp crimp decreases, as the warp straightens
    3. p₂ stays constant and only the warp crimp changes
    4. p₂ and the warp crimp both decrease, because the warp is shorter
    Show answer

    Answer: A — p₂ decreases as D ÷ sin θ₁ and the warp crimp increases

    With a straight weft p₂ = D ÷ sin θ₁, so a larger angle gives closer weft threads, and the crimp θ₁ sin θ₁ + cos θ₁ − 1 grows with θ₁ (about 12.8% at 30° and 21.5% at 40°). A steeper warp does not straighten, and p₂ is fixed only by D and θ₁, not independent of them.

  9. Which of the following are assumptions of Peirce’s basic circular-thread model of plain-woven cloth?

    1. Each thread has a circular cross-section that does not flatten
    2. The axis of each thread follows circular arcs joined by straight lines
    3. The threads flatten into elliptical sections where they cross under pressure
    4. Warp and weft are required to have the same diameter and the same spacing
    Show answer

    Answer: A — Each thread has a circular cross-section that does not flatten; B — The axis of each thread follows circular arcs joined by straight lines

    The basic model uses round, non-flattening threads and arc-plus-straight axes. Flattening into ellipses belongs to extended models that relax the circular-section assumption, and the equations are written with d₁, d₂, p₁, p₂ so unequal threads are allowed.

  10. Which of the following statements about plain weft-knitted fabric, in a fixed relaxed state, are correct?

    1. The courses per cm are inversely proportional to the loop length
    2. For a given yarn, a larger tightness factor means a smaller loop length and a tighter fabric
    3. For a given yarn, the mass per unit area is proportional to the loop length
    4. The stitch density per cm² is proportional to the loop length
    Show answer

    Answer: A — The courses per cm are inversely proportional to the loop length; B — For a given yarn, a larger tightness factor means a smaller loop length and a tighter fabric

    c = K_c ÷ l, and TF = √tex ÷ l rises when l falls. Mass per area is 0.1 K_s tex ÷ l, so it is inversely proportional to l, not proportional. Stitch density K_s ÷ l² falls as l grows.

  11. Which of the following statements about the geometry of a biaxial tubular braid are correct?

    1. tan θ = πD ÷ h, with θ measured from the axis and h the pitch of one yarn’s helix
    2. The yarn length in one turn is √[(πD)² + h²]
    3. At a fixed yarn length per turn, the volume enclosed keeps increasing as θ is raised towards 90°
    4. For fixed N, w and D the cover factor rises as the braid angle rises
    Show answer

    Answer: A — tan θ = πD ÷ h, with θ measured from the axis and h the pitch of one yarn’s helix; B — The yarn length in one turn is √[(πD)² + h²]; D — For fixed N, w and D the cover factor rises as the braid angle rises

    The tan θ relation and the turn length come from unrolling the helix, and the cover F = N w ÷ (2π D cos θ) rises with θ because cos θ falls. Volume is maximal at 54.7° and then falls as θ approaches 90° where the braid collapses to a flat ring of tiny height.

  12. In Peirce’s model a plain weave has a straight weft and the warp makes a maximum angle θ₁ = 40° with the cloth plane. Using crimp = θ₁ sin θ₁ + cos θ₁ − 1 (θ₁ in radians), where crimp is the thread length over the fabric length minus one, the warp crimp in per cent, to one decimal place, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 21.5

    θ₁ = 40° = 0.6981 rad. sin 40° = 0.6428 and cos 40° = 0.7660. Crimp = 0.6981 × 0.6428 + 0.7660 − 1 = 0.4488 + 0.7660 − 1 = 0.2148, that is 21.5%. Using degrees in place of radians is the usual slip.

  13. In a plain-woven cloth the warp yarn has diameter d₁ = 0.20 mm and the weft yarn has diameter d₂ = 0.30 mm, and the threads touch at the crossings as in Peirce’s model. The peak-to-peak crimp height of the warp is h₁ = 0.35 mm. Taking cloth thickness T = max (h₁ + d₁, h₂ + d₂), the thickness in mm, to two decimal places, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 0.55

    D = d₁ + d₂ = 0.50 mm, so h₂ = D − h₁ = 0.15 mm. Warp envelope = 0.35 + 0.20 = 0.55 mm; weft envelope = 0.15 + 0.30 = 0.45 mm. T = the greater = 0.55 mm. Adding all four numbers or using only the warp diameter are slips.

  14. A balanced plain weave is made of identical round yarns of 30 tex with yarn density ρ = 0.75 g/cm³. Take d = 0.0357 √(tex ÷ ρ) mm and the circular-thread jamming limit p_min = √3 d for each system. The maximum number of threads per cm in each direction, to the nearest whole number, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 26

    d = 0.0357 √(30 ÷ 0.75) = 0.0357 × 6.325 = 0.2258 mm. p_min = 1.732 × 0.2258 = 0.391 mm. Threads per cm = 10 ÷ 0.391 = 25.6, which is 26. Using p = d would give 44, which ignores the crimp geometry.

  15. A plain-woven fabric has 30 ends per cm of 20 tex warp with 8% warp crimp and 25 picks per cm of 25 tex weft with 5% weft crimp, both crimps defined on the fabric length. Ignoring size and finish, the mass per unit area in g/m², to one decimal place, is ____. (tex = mg per m)

    Numerical answer — type the value.

    Show answer

    Answer: 130.4

    Warp yarn per m² of cloth = 3000 ends × 1.08 = 3240 m, mass = 3240 × 20 mg = 64.8 g. Weft yarn per m² = 2500 picks × 1.05 = 2625 m, mass = 2625 × 25 mg = 65.625 g. Total = 130.425, which is 130.4 g/m². Ignoring crimp gives 125.0 g/m².

  16. A plain weft-knitted fabric is made from 25 tex yarn with a loop length l = 2.8 mm, and its Munden constants in the relaxed state are K_c = 5.0 and K_w = 4.0. Using areal density = 0.1 S l tex (g/m², with l in cm and S in loops per cm²) and S = K_c K_w ÷ l², the areal density in g/m², to one decimal place, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 178.6

    l = 0.28 cm. S = 20 ÷ 0.28² = 255.1 loops per cm². Areal density = 0.1 × 255.1 × 0.28 × 25 = 178.6 g/m² (equivalently 0.1 × 20 × 25 ÷ 0.28). Using l in mm in the formula would give a result ten times too large.

  17. A tubular braid of mean diameter D = 8.0 mm is made with each yarn circling the axis at f = 0.60 revolutions per second and the braid taken up at v = 24 mm/s. Using tan θ = π D f ÷ v, with θ measured from the braid axis, the braid angle in degrees, to one decimal place, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 32.1

    Pitch h = v ÷ f = 24 ÷ 0.60 = 40 mm. tan θ = π × 8.0 ÷ 40 = 0.6283, so θ = arctan 0.6283 = 32.1°. Measuring the angle from the circumferential direction would give 57.9° instead.