Friction, belts and screw jacks

In short

Friction is the resistance that acts along the surface of contact when one body slides or tends to slide on another. Engineering mechanics treats it as a quantified force (F = μN) and then applies it to machines that depend on friction to work — screw jacks, wedges, belts, ropes, brakes, clutches and bearings. The topic is scoring in first-year Engineering Mechanics, reappears in Theory of Machines and Design of Machine Elements, and is a standing GATE/ESE item (ratio of tensions, maximum power, self-locking condition). Interviewers at Tata Motors, Mahindra, L&T, Thermax, Kirloskar, Elecon, Fenner, ABB, BHEL and every plant-maintenance role ask why a jack holds its load, why a V-belt grips better than a flat belt and why belt power peaks and then falls with speed.

Explained

Dry (Coulomb) friction: laws, angles and types

• Static friction resists impending motion and rises to a limit; limiting friction F = μ_s N is that maximum; kinetic (dynamic) friction F = μ_k N acts once sliding starts, and μ_k < μ_s.
• Laws of dry friction: (1) friction opposes the direction of relative motion or its tendency; (2) F is proportional to the normal reaction N; (3) limiting F is independent of the apparent area of contact; (4) it depends on the nature and roughness of the surfaces; (5) kinetic friction is nearly independent of sliding speed at moderate speeds.
• Coefficient of friction μ = F/N, dimensionless. Angle of friction φ = tan⁻¹μ — the angle the resultant of F and N makes with the normal.
• Angle of repose is the maximum inclination of a plane on which a body just rests without sliding; it equals the angle of friction (tan θ = μ). This is how μ is measured in the lab tilting-plane experiment.
• Cone of friction: rotating the resultant about the normal sweeps a cone of semi-angle φ. If the resultant applied force lies inside the cone the body cannot slide — no matter how large the force.
• Types: dry/solid friction (between surfaces), fluid friction (viscous, in a lubricated film — proportional to velocity, not to N), film/boundary friction (thin layer, partly dry), and rolling resistance (deformation of wheel and surface; coefficient 0.001–0.02, an order smaller than sliding friction — the reason rolling-element bearings replaced plain ones).
• Typical values (indicative; vary with finish, lubrication, temperature): steel on steel dry 0.15–0.30, lubricated 0.05–0.10; leather belt on cast iron 0.25–0.35; rubber on dry concrete 0.6–0.9; brake lining on steel 0.30–0.40; PTFE on steel ≈ 0.04.
• Friction is a non-conservative force: the work it does becomes heat (W_f = F × distance), which is why brake drums and jack threads get hot.

Bodies on planes, ladders and wedges

• Body of weight W on a plane inclined at α, pulled by force P at angle θ to the plane, moving up:
  • P parallel to plane: P = W(sin α + μ cos α).
  • P horizontal: P = W tan(α + φ).
  • General: P = W sin(α + φ)/cos(θ − φ); least force occurs at θ = φ, giving P_min = W sin(α + φ).
• Moving down (or holding it): replace φ by −φ → P = W tan(α − φ). If α < φ the body will not slide on its own (it is self-locking on the plane); if α > φ a holding force is needed.
• Ladder friction: a uniform ladder of weight W leaning against a smooth vertical wall with a rough floor is in equilibrium if tan θ ≥ 1/(2μ), where θ is the angle with the floor. A man of weight W₁ at a distance a along a ladder of length L adds a moment W₁a cos θ — the standard exam variant. If the wall is also rough, include its friction force (vertical, upward) in both force and moment equations.
• Wedge friction: used to lift heavy loads through small distances (machine levelling, erection of transformers, press tools). Draw a free body for the wedge and another for the load block; friction acts on both faces of the wedge and opposes its motion; solve by force triangles (Lami's theorem) or by resolving. A wedge of included angle α is self-retaining when α < 2φ (friction on both faces) — practical wedges use small angles so they stay put.
• Screw and wedge are the same mechanism: a screw thread is an inclined plane (a wedge) wrapped round a cylinder.

Power screws and the screw jack

• A power screw converts rotation into axial thrust: screw jacks, vices, C-clamps, presses, lead screws of lathes, valve actuators, tail-stock spindles.
• Thread forms: square thread (highest efficiency, no radial bursting force, used in jacks and presses); Acme (29° included, half-angle 14.5°) and ISO metric trapezoidal (30° included, half-angle 15°) — stronger, and allow a split nut for wear take-up; buttress (one-direction load). V-threads are not meant for power transmission because their wedging raises friction.
• Geometry: pitch p = axial distance between adjacent threads; lead l = p × number of starts; mean diameter d_m = (d_o + d_i)/2 ≈ d_o − p/2 for square threads; helix angle tan α = l/(π d_m).
• Operation: unwrapping one turn of the thread gives an inclined plane of inclination α carrying load W, pushed horizontally by force P_t = W tan(α + φ) at the mean radius. Hence the torque to raise T = W (d_m/2) tan(α + φ), and to lower T = W (d_m/2) tan(φ − α).
• Effort at the handle: P = T/L, where L is the handle (lever) radius — this is the mechanical advantage that lets one person lift a truck.
• Self-locking: the screw holds the load without a brake when φ ≥ α (equivalently μ ≥ tan α, or η ≤ 50%). A jack must be self-locking; a lead screw meant for fast travel need not be. Overhauling (self-lowering) screws need a brake or a non-reversible worm.
• Efficiency η = tan α / tan(α + φ); it is independent of the load. η is maximum at α = 45° − φ/2, with η_max = (1 − sin φ)/(1 + sin φ). A self-locking screw can never exceed 50% efficiency — the price paid for safety.
• Collar/thrust friction: the load sits on a collar or pad of mean radius R_c, adding T_c = μ_c W R_c (uniform wear) which does no useful work and cuts overall efficiency sharply. Good jacks use a hardened pad or a ball/thrust bearing to cut μ_c.
• For V-threads use a virtual coefficient μ′ = μ/cos β (β = half thread angle) in place of μ.
• Types of jack: simple screw jack; differential screw jack (two threads of pitch p₁ and p₂ on the same spindle in series — effective lead = p₁ − p₂, so velocity ratio = 2πL/(p₁ − p₂), very large movement ratio in a short body); compound screw jack (threads of opposite hand, effective lead = p₁ + p₂); bottle/worm-geared jack (bevel or worm gearing between handle and screw, used in railway and bridge jacking); mechanical toe jack. A hydraulic jack is not a screw jack — it works on Pascal's law, is not self-locking by thread, and needs a non-return valve.

Belt and rope drives: geometry and ratio of tensions

• A belt transmits power by friction between belt and pulley. Tight side tension T₁, slack side T₂; the driving force is (T₁ − T₂).
• Types: flat belt (leather, canvas, rubber, nylon-core), V-belt (wedge grip, short centres, multiple belts, classical sections Z, A, B, C, D, E per IS 2494), timing/toothed belt (positive, no slip, used in camshaft drives and 3-D printers), round/rope drives, and flat/multi-rib (Poly-V) belts.
• Open belt (pulleys rotate the same way) vs crossed belt (opposite ways, more wear at the crossing). Idler/jockey pulley increases the arc of contact and takes up slack; compound drive uses intermediate shafts.
• Angle of lap (contact) θ, in radians, for centre distance x and radii r₁ > r₂: sin α = (r₁ − r₂)/x (open) or (r₁ + r₂)/x (crossed); θ_small = π − 2α, θ_large = π + 2α (open); θ = π + 2α for both pulleys (crossed). Always use the smaller θ (the smaller pulley) in design — it slips first.
• Velocity ratio N₂/N₁ = d₁/d₂; including belt thickness t, N₂/N₁ = (d₁ + t)/(d₂ + t); with total percentage slip s, multiply by (1 − s/100).
• Slip is gross relative motion from insufficient friction (power loss, lost ratio); creep is the small elastic contraction/extension of the belt as it moves from tight to slack side, giving N₂/N₁ = (d₁/d₂)·(E + √σ₂)/(E + √σ₁).
• Crowning of flat pulleys keeps the belt centred; a flat belt climbs to the tightest (largest) part.
• V-belt advantage: the wedge action multiplies the normal reaction, so the effective exponent becomes μθ/sin β (β = half groove angle, typically 17°–19°; grooves are cut slightly narrower than the belt angle so the belt wedges in as it bends).
• Limitations of belts: slip and creep make them non-positive, they need initial tension and periodic re-tensioning, they are speed- and temperature-limited and they stretch; chains and gears are used where exact ratio or high torque matters. Advantages: cheap, quiet, absorb shock, tolerate misalignment and large centre distances, and act as an overload safety by slipping.

Power, centrifugal tension and maximum power

• Belt speed v = πdN/60. Power P = (T₁ − T₂)v. Torque on a pulley = (T₁ − T₂)r.
• Centrifugal tension T_c = m v² (m = mass per metre of belt) is caused by the belt's own mass turning the pulley; it adds equally to both sides, so it does not help transmit power but it does eat into the allowable tension: T_max = σ·b·t = T₁ + T_c.
• The ratio equation uses only the friction-carrying parts: T₁/T₂ = e^μθ.
• Maximum power occurs when T_c = T_max/3, i.e. v = √(T_max/3m); beyond this speed the belt chokes on its own centrifugal tension and power falls.
• Initial tension T₀ = (T₁ + T₂ + 2T_c)/2 (assuming the belt material obeys Hooke's law and the total length is constant). Too little → slip; too much → bearing load and short belt life.
• Rope brake / band brake / block brake: the same friction relation gives braking torque = (T₁ − T₂)r, with T₁/T₂ = e^μθ for a band; for a block brake, braking torque = μNr. A differential band brake can be made self-energising (and must be checked for self-locking).

Pivot, collar, bearing and clutch friction

• Flat pivot (shaft end on a flat bearing): T = (2/3)μWR (uniform pressure).
• Flat collar / single-plate clutch, radii R and r:
  • Uniform pressure: T = (2/3)μW(R³ − r³)/(R² − r²).
  • Uniform wear (used for worn clutches and brakes): T = μW(R + r)/2 = μWR_m.
  • Multiply by the number of pairs of contacting surfaces (a single-plate clutch with friction on both faces has 2).
• Conical pivot / cone clutch: T = (2/3)μWR·cosec α, the cone angle multiplying the torque — the same wedge idea as the V-belt.
• Journal bearing friction is modelled as a friction circle of radius r sin φ; friction torque ≈ μWr.

Standards, codes and Indian practice

• BIS standards (check the current edition — BIS revises and sometimes withdraws numbers): IS 2494 — endless V-belts for industrial drives (sections and dimensions); IS 3142 — V-grooved pulleys; IS 1691 — flat transmission pulleys; IS 1891 — rubber conveyor and elevator belting; IS 4694 — square threads; IS 7008 — ISO metric trapezoidal screw threads.
• Factories Act, 1948 (repealed from 21 November 2025 by the OSH Code, 2020, whose rules and the saved State Factories Rules carry these duties): s.21 required secure fencing of all transmission machinery — belts, pulleys and couplings must be guarded; s.24 required striking gear or another efficient mechanical appliance to move driving belts to and from loose pulleys, and belts not in use must not ride on shafting in motion; s.29 required every lifting machine, chain, rope and lifting tackle (cranes, winches, pulley blocks) to be of good construction, tested, marked with its safe working load, and thoroughly examined at prescribed intervals; s.28 covers hoists and lifts.
• Design practice: apply a service factor (shock, hours of running, idler use) and an arc-of-contact factor to the catalogue power per belt from IS 2494 tables; the e^μθ figure is an idealised upper bound.
• Belt power transmission in a plant is also an energy-efficiency item — BEE's energy-audit practice credits replacing worn V-belts with flat/Poly-V or cogged belts, which cut slip losses by a few per cent.

Key points

• Laws of friction — F ∝ N, independent of apparent contact area, static limit > kinetic value.
• μ and φ — μ = tan φ; the resultant reaction leans φ from the normal at impending slip.
• Angle of repose — equals the angle of friction; the tilting-plane method of measuring μ.
• Cone of friction — a force whose line lies inside it can never cause sliding, however large.
• Least force on a plane — applied at φ to the plane, P_min = W sin(α + φ).
• Self-locking on a plane — a body stays put if the inclination α ≤ φ.
• Helix angle — tan α = lead/(π d_m); lead = pitch × number of starts.
• Screw torque — T = W(d_m/2)tan(α + φ) to raise, tan(φ − α) to lower.
• Self-locking screw — φ ≥ α, equivalently efficiency ≤ 50%; mandatory for a jack.
• Maximum screw efficiency — (1 − sin φ)/(1 + sin φ) at α = 45° − φ/2; efficiency is load-independent.
• Collar friction — pure loss, μ_cWR_c; remove it with a thrust bearing.
• Ratio of tensions — T₁/T₂ = e^μθ (flat), e^{μθ cosec β} (V-belt); θ in radians, smaller pulley governs.
• Centrifugal tension — T_c = mv²; adds to both sides, transmits no power, limits T₁ since T_max = T₁ + T_c.
• Maximum power — at T_c = T_max/3, v = √(T_max/3m); faster is not more powerful.
• Initial tension — T₀ = (T₁ + T₂ + 2T_c)/2.
• Slip vs creep — slip is gross sliding (friction insufficient); creep is elastic stretch difference between the two sides.
• Clutch/collar torque — uniform wear T = μW(R + r)/2; uniform pressure T = (2/3)μW(R³ − r³)/(R² − r²).
• Rolling resistance — coefficient 0.001–0.02, far below sliding friction; the basis of wheels and ball bearings.

Formulas and examples

Example 1 — block on an inclined plane (all three cases). W = 500 N, α = 20°, μ = 0.3 → φ = tan⁻¹0.3 = 16.70°.

• Force parallel to plane: P = 500(sin20° + 0.3 cos20°) = 500(0.3420 + 0.2819) = 312 N.
• Horizontal force: P = 500 tan(36.70°) = 500 × 0.7454 = 373 N.
• Least force, applied at 16.70° to the plane: P = 500 sin(36.70°) = 500 × 0.5976 = 299 N — the smallest of the three, which is why a rope is pulled slightly upward, not along the surface.
• Since α (20°) > φ (16.7°), the block slides down unaided; force to hold it = 500 tan(3.30°) = 28.8 N.

Example 2 — screw jack with collar friction. Single-start square thread, outside diameter 50 mm, pitch 10 mm, μ = 0.15 (φ = 8.53°), load W = 20 kN, handle length 600 mm, collar mean radius 40 mm, μ_c = 0.12.

1. d_m = 50 − 10/2 = 45 mm; tan α = 10/(π × 45) = 0.07075 → α = 4.05°. Since φ = 8.53° > α, the jack is self-locking.
2. Screw torque to raise = 20 000 × 0.0225 × (0.07075 + 0.15)/(1 − 0.15 × 0.07075) = 450 × 0.22311 = 100.4 N·m.
3. Screw efficiency η = 0.07075/0.22311 = 31.7% (check: W·l/2πT = 20 000 × 0.010/(2π × 100.4) = 0.317 ✓).
4. Torque to lower = 450 × tan(8.53° − 4.05°) = 450 × 0.07843 = 35.3 N·m — positive, confirming self-locking.
5. Collar torque = 0.12 × 20 000 × 0.040 = 96 N·m. Total = 196.4 N·m → effort at handle P = 196.4/0.6 = 327 N.
6. Overall efficiency = 20 000 × 0.010/(2π × 196.4) = 16.2%. VR = 2π × 600/10 = 377; MA = 20 000/327 = 61.2.
7. Best possible efficiency for this thread friction = (1 − sin 8.53°)/(1 + sin 8.53°) = 0.8517/1.1483 = 74.2% at α = 40.7° — but such a lead would not be self-locking, so a jack deliberately sacrifices it.

Example 3 — flat belt power and maximum power. Belt 100 mm × 8 mm, density 1 000 kg/m³, permissible stress 2.5 MPa; pulley 400 mm at 600 rpm; μ = 0.3, angle of lap 165°.

1. v = π × 0.4 × 600/60 = 12.57 m/s; m = 1 000 × 0.1 × 0.008 = 0.8 kg/m.
2. T_max = 2.5 × 10⁶ × 0.1 × 0.008 = 2 000 N; T_c = 0.8 × 12.57² = 126.3 N; T₁ = 2 000 − 126.3 = 1 873.7 N.
3. θ = 165° = 2.880 rad; μθ = 0.864; e^0.864 = 2.373 → T₂ = 1 873.7/2.373 = 789.8 N.
4. P = (1 873.7 − 789.8) × 12.57 = 13.6 kW. Initial tension T₀ = (1 873.7 + 789.8 + 252.6)/2 = 1 458 N.
5. Maximum-power speed v = √(2 000/(3 × 0.8)) = 28.87 m/s (≈ 1 379 rpm on this pulley), where T_c = 666.7, T₁ = 1 333.3, T₂ = 562.0 → P_max = 771.3 × 28.87 = 22.3 kW. The drive is running well below its optimum speed.

Example 4 — why a V-belt grips better. Groove half-angle β = 19°, μ = 0.25, θ = 160° = 2.793 rad.

• Flat belt: T₁/T₂ = e^{0.25 × 2.793} = e^0.698 = 2.01.
• V-belt: exponent = 0.698/sin19° = 0.698/0.3256 = 2.144 → T₁/T₂ = e^2.144 = 8.54, i.e. 4.2 times the flat-belt tension ratio, which raises the usable tension difference for the same tight-side tension by about 1.8 times (0.883 T₁ against 0.503 T₁).
• With T_max = 800 N and m = 0.19 kg/m at v = 15 m/s: T_c = 0.19 × 225 = 42.8 N, T₁ = 757.2 N, T₂ = 88.7 N → P = 668.5 × 15 = 10.0 kW per belt (an upper bound; IS 2494-based catalogue ratings are lower after arc-of-contact, length and service factors).

Example 5 — belt length, lap angle and velocity ratio. d₁ = 600 mm, d₂ = 300 mm, centre distance x = 1.5 m, N₁ = 500 rpm.

1. Open belt: sin α = (0.30 − 0.15)/1.5 = 0.1 → α = 5.74°; θ_small = 180° − 11.48° = 168.5° (2.941 rad), θ_large = 191.5°.
2. L_open = π(0.45) + 2(1.5) + (0.15)²/1.5 = 1.414 + 3 + 0.015 = 4.43 m.
3. Crossed belt: α = sin⁻¹(0.45/1.5) = 17.46° → θ = 214.9°; L = 1.414 + 3 + 0.2025/1.5 = 4.55 m (longer, and both pulleys get the larger lap).
4. Ideal N₂ = 500 × 600/300 = 1 000 rpm. With belt thickness 5 mm and 3% total slip: N₂ = 500 × (605/305) × 0.97 = 962 rpm — a 3.8% shortfall, which is why belts are not used where the ratio must be exact.

Where it is used

• Lifting and erection: bottle and toe screw jacks for vehicles, railway coach jacking, levelling of turbines, transformers and machine tools; screw jacks under formwork props and pipeline supports.
• Machine tools and actuators: lead screws of lathes and milling machines, bench vices, C-clamps, screw presses, valve stems and gate-valve actuators, electro-mechanical linear actuators.
• Belt drives everywhere in Indian industry: V-belt drives on Kirloskar/Crompton pumps, Elgi and Atlas Copco compressors, Fenner and Gates belts in textile mills, sugar and cement plants; flat and Poly-V belts on alternators and car accessory drives, rubber V-belts in two-wheeler CVTs; conveyor belting (IS 1891) in coal handling and mining.
• Brakes and clutches: band and shoe brakes on cranes and hoists, railway brake blocks, single-plate dry clutches in cars and multi-plate wet clutches in motorcycles and tractors.
• Traction and civil applications: tyre–road and wheel–rail friction limits acceleration and braking; friction piles, retaining-wall and soil friction angle in geotechnical design; friction grip (HSFG) bolts in steel structures.
• Job roles: maintenance and reliability engineers (belt tensioning, alignment, slip and energy audits), design engineers (power-screw and brake sizing), plant safety officers (OSH Code guarding — formerly the Factories Act — and SWL marking of lifting machines and tackle).

Interview questions

• Q: Why is the angle of repose equal to the angle of friction? A: On a plane at the limiting inclination, W sin θ = μW cos θ, so tan θ = μ = tan φ.
• Q: Does friction depend on the contact area? A: Not on the apparent area — F = μN. Real contact area adjusts with pressure; very large or very small areas, high pressures, soft materials and tyres do deviate from this ideal.
• Q: What makes a screw jack self-locking, and what does it cost? A: The friction angle must be at least the helix angle (φ ≥ α); the cost is efficiency, which can then never exceed 50%.
• Q: Is the efficiency of a screw jack affected by the load? A: No — η = tan α/tan(α + φ) depends only on the helix and friction angles, so it is a geometry-and-friction property, not a load property.
• Q: Why are square threads used in jacks rather than V-threads? A: No wedging means lower friction and higher efficiency, and no radial bursting force on the nut; V-thread friction must be computed with μ′ = μ/cos β.
• Q: Why does a V-belt transmit more power than a flat belt of the same tension? A: The groove wedges the belt, so the normal reaction is amplified by 1/sin β and the tension ratio becomes e^{μθ/sin β} instead of e^μθ.
• Q: Does centrifugal tension help transmit power? A: No. It adds equally to both sides so it cancels from (T₁ − T₂), but it consumes part of the allowable tension and therefore reduces the power the belt can carry.
• Q: Why does belt power first rise and then fall with speed? A: Power = (T₁ − T₂)v rises with v, but T_c = mv² eats into T₁; the maximum is at T_c = T_max/3, v = √(T_max/3m).
• Q: Which pulley's angle of lap do you use in design? A: The smaller pulley's, because its smaller arc of contact gives the lower tension ratio and it slips first.
• Q: Difference between slip and creep? A: Slip is gross sliding between belt and pulley when friction is inadequate; creep is the small loss of speed caused by the belt's elastic stretch differing on the tight and slack sides — it occurs even with no slip.
• Q: Why is rolling resistance so much lower than sliding friction? A: There is no sliding at the contact; the loss comes only from hysteresis in deforming the wheel and the surface, giving coefficients of 0.001–0.02.
• Q: A hydraulic jack and a screw jack both lift a car — which one can fail by leaking down, and why? A: The hydraulic jack, because it holds load by fluid pressure behind a valve and seal; a self-locking screw jack is held by thread friction, which does not leak.

Common mistakes

• Using θ in degrees in e^μθ, or taking the larger pulley's lap angle instead of the smaller one.
• Writing T₁/T₂ = e^μθ with T₁ = T_max — the centrifugal part must be subtracted first: T₁ = T_max − T_c.
• Assuming the direction of friction instead of deducing it: friction always opposes the *impending* relative motion, which reverses between raising and lowering.
• In screw problems, using the outside diameter or the pitch in place of the mean diameter and the lead (a two-start thread has lead = 2 × pitch).
• Forgetting collar/thrust friction, which can halve the overall efficiency of a jack and is the usual reason a measured effort exceeds the calculated one.
• Treating limiting friction as if it always acts — before slipping, friction is only as large as the applied force needs it to be (check the equilibrium value against μN).
• Confusing the self-locking condition for a plane (α ≤ φ) with the maximum-efficiency condition (α = 45° − φ/2), and quoting both for the same screw.
• Adding the two friction forces on a wedge as if they acted on the same body, instead of drawing separate free bodies for wedge and load.

Related topics

Engineering mechanics: statics and dynamics · Design of machine elements: power screws, bolted joints and springs · Theory of machines: brakes, clutches, dynamometers and flywheels · Gears, chains and drive selection · Tribology, lubrication and bearings · Strength of materials: stress, strain and factor of safety