Friction
Dry friction is the tangential resistance that develops where two unlubricated surfaces meet. This chapter builds the laws governing that resistance and applies them to the machines that depend on it — wedges, screws, belts, bearings, and rolling wheels.
Learning objectives
- Introduce dry friction and analyze the equilibrium of rigid bodies subjected to frictional forces.
- Apply frictional analysis to specific machine elements: wedges, screws, belts, and bearings.
- Investigate the concept of rolling resistance.
Characteristics of Dry Friction
Friction always acts tangent to the contacting surfaces and points so as to oppose the existing or impending relative motion between them. Dry (Coulomb) friction is the case where no lubricating fluid separates the surfaces.
Dry (Coulomb) Friction
The tangential resisting force that arises between two solid surfaces in contact with no intervening lubricant. It originates in the interlocking of microscopic surface irregularities, each of which contributes both a normal and a frictional reaction component.
From distributed contact to a resultant
A block pulled horizontally rests on countless contact points, each supplying a small normal force and a small frictional force. Summing these gives a single normal resultant N and frictional resultant F. The normal resultant acts at an offset x from the load line so that it can balance the tipping moment of the applied force.
Fig. 8.1Resultant free-body diagram of a block on a rough surface. The normal resultant N shifts a distance x to counter the tipping moment produced by the applied force P acting at height h.
Impending motion and the limiting force
As the applied load grows, the static friction force rises to match it — up to a ceiling. At that ceiling the body is on the verge of sliding, and the friction force takes its maximum static value.
Limiting Static Frictional Force
The largest friction force that can develop before sliding begins. Experiments show it is directly proportional to the normal resultant. At this point the body is in unstable equilibrium: any further increase in load starts motion.
Coefficient of Static Friction (μs)
The dimensionless constant of proportionality between the limiting friction force and the normal force. Its value depends on the materials and surface condition, so it should be treated as an experimental estimate rather than an exact constant.
Angle of Static Friction (φs)
The angle between the resultant contact reaction and the normal direction when sliding impends. It relates directly to the coefficient of friction and is independent of the magnitude of the normal force.
| Contact materials | Coefficient of static friction (μs) |
|---|---|
| Metal on ice | 0.03 – 0.05 |
| Wood on wood | 0.30 – 0.70 |
| Leather on wood | 0.20 – 0.50 |
| Leather on metal | 0.30 – 0.60 |
| Copper on copper | 0.74 – 1.21 |
Kinetic friction
Once sliding begins, the resisting force drops to a lower, roughly constant kinetic value. The block then rides over the surface peaks, and continued breakdown of those peaks sustains the kinetic force.
Kinetic coefficients run roughly 25% below the static values, and the kinetic friction angle satisfies \( \phi_s \ge \phi_k \).
For bodies subjected to dry friction, experiment establishes the following rules:
1. The friction force acts tangent to the contact surface, opposing motion or the tendency toward motion.
2. The maximum static friction force is independent of the contact area, provided the normal pressure is neither extremely low nor high enough to crush the surfaces.
3. The maximum static friction force generally exceeds the kinetic friction force for the same pair of surfaces.
4. When slipping is about to occur, \( F_s = \mu_s N \). When slipping is occurring, \( F_k = \mu_k N \).
Physical meaning
Static friction is not automatically \(\mu_sN\): it supplies only the tangential reaction equilibrium demands, up to that ceiling. A normal force is perpendicular to contact; friction reverses if the predicted relative motion reverses.
Sign convention matters more than memorizing formulas here. When you don't yet know whether motion impends, leave F as an unknown and let equilibrium reveal its sign. Only switch to F = μN once you've confirmed (or are told) that the surface is at the verge of slipping. Mixing these two up is the single most common mistake I see in friction problems.
Problems Involving Dry Friction
A body in equilibrium under a force system that includes friction must satisfy both the equilibrium equations and the laws governing the friction force. Classify the problem by comparing the count of unknowns with the count of available equations.
No Apparent Impending Motion
A pure equilibrium problem: unknowns equal the equilibrium equations. Solve for the friction forces, then verify each satisfies \( F \le \mu_s N \). If any does not, the assumption of equilibrium fails and slipping occurs.
Impending Motion at All Contact Points
Unknowns equal the equilibrium equations plus one friction equation per contact. With motion impending everywhere, \( F = \mu_s N \) holds at every contact simultaneously.
Impending Motion at Some Contact Points
Several modes of motion are possible. Each candidate (slip here but not there) is solved separately; the physically correct mode is the one requiring the smallest driving force.
When F is an unknown equilibrium force satisfying \( F < \mu_s N \), its direction may be assumed freely; a negative result simply reverses the assumed sense.
When the friction equation \( F = \mu N \) is used, this freedom is lost. Because that equation relates only the magnitudes of two perpendicular vectors, F must be drawn in its correct physical sense from the outset.
Problem-solving strategy
- Isolate a definite body and predict relative motion at each rough interface.
- Draw normal and tangential reactions using tangent-normal axes where helpful.
- Apply equilibrium with unknown friction unless slip is explicitly impending.
- Impose \(|F|=\mu N\) only at limiting contacts and compare plausible modes.
- Audit contact capacity, tipping, torque direction, and unit consistency.
Procedure for analysis
Draw the free-body diagram and show friction forces as unknowns unless slipping is stated. Count unknowns against equilibrium equations; if there are too few equations, introduce friction equations at the appropriate contacts. Then solve, applying conditional (tipping) equations where a body might overturn.
Wedges
A wedge is a simple machine that converts an applied force into much larger forces acting nearly at right angles to it, and is used to lift or finely adjust heavy loads.
Wedge
An inclined plane driven beneath or between loads. A small force along the wedge produces a large normal force across it. Analysis uses force equilibrium on both the wedge and the block, plus a friction equation at each contact; moment equations are usually unnecessary since neither body tips.
Self-Locking Wedge
A wedge that stays in place after the driving force is removed, held only by friction. This occurs when the friction coefficients are large enough (or the wedge angle small enough) that the friction forces alone balance the load.
Fig. 8.2A wedge of angle θ is driven by force P to raise the block of weight W against a smooth guide. Friction acts on each of the wedge's faces to oppose its motion.
Direction discipline is everything with wedges. Sketch the impending motion first, then draw every friction force opposing it. If the block is being lowered instead of raised, flip all friction arrows together — not one at a time. Forget this and your equilibrium equations will quietly give you a self-locking answer when the wedge actually slides.
Frictional Forces on Screws
A square-threaded screw is an inclined plane wrapped around a cylinder. Unwinding one turn of the thread reveals a ramp whose rise is the lead and whose run is the mean circumference.
Lead and Lead Angle
The lead l is the axial distance the screw advances in one revolution. The lead angle θ is the slope of the unwrapped thread, found from the lead and the mean thread radius.
Treating the unwrapped thread as a block on an incline driven by the horizontal force M∕r (from the applied torque) leads to the moment relations below.
Self-Locking Screw
A screw that holds its axial load with the torque removed. This requires the friction angle to meet or exceed the lead angle (φs ≥ θ); when equal, the screw is on the verge of winding down on its own.
Practical check for self-locking: \( \phi_s \ge \theta \) is the same as \( \mu_s \ge \tan\theta = l/2\pi r \). Standard fastening screws have small leads precisely so they stay put — that's why a bolt doesn't spin loose under load. Power screws (jacks, vises) trade some of that locking margin for faster travel.
Frictional Forces on Flat Belts
When a belt wraps a fixed curved surface, friction along the contact arc lets one belt tension hold against a much larger one. Designing belt drives and band brakes requires relating those two tensions.
Belt Friction Setup
A belt contacts a drum over an angle β (in radians). The tension T2 driving the belt in the direction of impending motion always exceeds the resisting tension T1 because friction along the arc adds up. Integrating equilibrium over a differential element yields the governing relation.
Fig. 8.3A flat belt over a fixed drum. The contact angle β is measured in radians; the larger tension T2 acts in the direction of the belt's impending motion.
Key Properties of the Belt Equation
The tension ratio depends only on the friction coefficient and the wrap angle β — not on the drum radius. Adding wraps multiplies β (and therefore the holding capacity) rapidly, which is why a few turns around a post can restrain an enormous load.
Collar Bearings, Pivot Bearings & Disks
Pivot and collar bearings support an axial load on a rotating shaft. Without lubrication, dry-friction laws give the moment needed to turn the shaft against the distributed friction over the contact ring.
Collar / Pivot Bearing Friction
The axial load spreads over an annular contact area as a (uniform) pressure. Each differential ring element carries a friction force; integrating its moment about the shaft axis gives the total turning moment. The result assumes uniform pressure on a new, evenly supported bearing.
Fig. 8.4Collar bearing carrying axial load P and turning moment M. Uniform contact pressure p acts over the annulus between inner radius R1 and outer radius R2.
Equations 8–7 and 8–8 only hold for uniform pressure. The moment for constant-speed rotation uses μk in place of μs. If the pressure varies (worn bearing, linear or parabolic distribution), you must re-derive: write p(r), form dF = μp·dA, and integrate r·dF from scratch. a fresh distributed-pressure derivation is required.
Frictional Forces on Journal Bearings
A journal bearing supports a shaft under lateral load. As the shaft turns, its contact point climbs the bearing wall until slipping fixes the location of the reaction — defining a small friction circle the reaction is always tangent to.
Friction Circle
As the shaft rotates, the bearing reaction R (equal and opposite to the load P) shifts to a slipping point and stays tangent to a circle of radius rf = r sinφk about the shaft center. The turning moment equals R times this radius.
For small φk, sinφk ≈ tanφk ≈ μk, which gives the convenient approximate form. Smaller bearing radii reduce friction, but plain journal bearings wear; ball or roller bearings are substituted for long service.
Rolling Resistance
No surface is perfectly rigid. A rolling cylinder deforms the surface ahead of it, so the normal reaction shifts forward and tilts — producing a resistance that a horizontal force must overcome.
Rolling Resistance
The motion resistance of a rolling body caused mainly by localized deformation of the contacting materials (with smaller contributions from adhesion and micro-sliding). Because the front material resists more than the rear restores, a driving force is needed to keep rolling at constant speed.
Coefficient of Rolling Resistance (a)
The forward offset of the resultant normal force, expressed as a length. Unlike a friction coefficient it carries units of distance; e.g. about 0.5 mm for a steel wheel on a steel rail. It is hard to measure precisely because it depends on speed, elasticity, and surface finish.
Fig. 8.5Simplified model of rolling resistance. The resultant normal force N acts at point A, offset a distance a ahead of the wheel center, opposing the driving force P.
Because Wa∕r is usually far smaller than μkW, rolling a load takes much less force than sliding it — the basic reason rollers and ball bearings reduce friction between moving parts.
Watch the units: a is a length, not a ratio. When you see "coefficient of rolling resistance = 0.5 mm," it's literally the forward shift of the normal force. The clean rolling-test relation — a = r·tanθ at the constant-velocity angle — is the fastest way to extract it in the lab.
Recognition strategy
This is a particle on an incline with no stated impending motion, so solve for the static-friction demand before testing the limit.
Given information
- \(m=32\ \text{kg}\), \(\theta=16^\circ\), and dry static coefficient \(\mu_s=0.34\).
Model and interpretation
Use axes tangent and normal to the ramp. Weight tends to move the case down-ramp, so friction on the case acts up-ramp.
Verification
Units: Weight, normal force, and friction are all newtons; \(\mu_s\), sine, and cosine are dimensionless, so no incompatible units enter the comparison.
Signs and direction: The gravitational tangent component points down the ramp, therefore the positive calculated resisting force must physically point up the ramp.
Plausibility: Friction demand would vanish on a level floor and rise with angle; 86.5 N below 102.6 N is consistent with static equilibrium.
Engineering interpretation
The 16.1 N margin is modest; vibration or contamination can lower the real coefficient, so a transported instrument should still be restrained rather than relying on friction alone.
Recognition strategy
The capstan relation applies because a flexible band is at impending slip on a fixed curved drum; tension ratio, then torque difference, is needed.
Given information
- \(r=0.18\ \text{m}\), \(T_1=120\ \text{N}\), \(\mu_s=0.28\), and \(\beta=210^\circ\).
Model and interpretation
Take \(T_2\) as the tight side in the impending drum-rotation direction. Neglect band stiffness and assume uniform coefficient over the wrap.
Verification
Units: The exponential exponent is dimensionless because radians are dimensionless; the tension difference times radius correctly produces N·m.
Signs and direction: The tight side is defined by impending relative belt motion, so its larger tension produces a torque opposing the specified drum rotation.
Plausibility: With zero wrap the tensions would match and torque vanish; a 210° wrap giving a 2.79 ratio is physically credible for \(\mu=0.28\).
Engineering interpretation
More wrap increases holding capacity exponentially, whereas increasing drum radius increases torque linearly; lining wear and heat can change \(\mu\), requiring brake rating checks.
Recognition strategy
A square screw unwraps into an inclined plane. Raising requires the lead angle plus friction angle, then the locking criterion compares those angles.
Given information
- \(r=0.012\ \text{m}\), \(l=0.004\ \text{m/rev}\), \(W=18\,000\ \text{N}\), \(\mu=0.15\).
Model and interpretation
Assume quasi-static raising, square threads, and load centered on the screw. Collar torque, thread deformation, and dynamic effects are excluded.
Verification
Units: Radius times axial force gives N·m and tangent is dimensionless; lead and circumference were both converted to metres before their ratio.
Signs and direction: The positive torque is in the raising direction; both the slope and friction angle oppose the assumed upward motion rather than assist it.
Plausibility: Since \(\phi=8.53^\circ\) exceeds \(\theta=3.04^\circ\), unloading does not make the ideal screw back-drive, matching the locking conclusion.
Engineering interpretation
Real jacks require collar friction, strength, buckling, lubrication state, wear, and safety-factor review; self-locking alone must not be treated as a personnel-lifting safety device.
Chapter recap
- Use a friction inequality for static contact and equality only at the verified limit.
- Choose friction directions from relative motion and test competing slip modes.
- For belts use radians; for screws compare friction and lead angles; for rolling resistance retain its length unit.
- Before moving on: draw a contact FBD, solve equilibrium, test capacity, and state the applicable idealization.