Friction

Until now, surfaces in contact have mostly been treated as smooth, able to push only perpendicular to themselves. Real surfaces also resist sliding, and that resistance, friction, is everywhere in engineering. Sometimes it is wanted: brakes, clutches, belt drives, bolted joints that grip by friction, nails and screws, tires on roads, footings that must not slide, and workers who must not slip. Sometimes it is a loss to be minimized: in bearings, gears and machine slides. And sometimes it decides whether a system works at all: whether a wedge stays in place, whether a screw jack holds its load, whether a ladder slips. This lesson covers the laws of dry friction, problems in which an object may slide or tip, wedges, screws and belt friction, and the practical uses of each.

The laws of dry friction

When one dry surface rests on another, the contact force has two components: the normal force N, perpendicular to the surfaces, and the friction force F, along them, which opposes any tendency to slide. Experiments, first systematized by Leonardo da Vinci and later by Guillaume Amontons and Charles-Augustin de Coulomb, show that:

• While there is no sliding, the friction force is whatever is needed for equilibrium, up to a limit: \(F \le \mu_s N\), where \(\mu_s\) is the coefficient of static friction.
• When sliding is about to start (impending motion), \(F = \mu_s N\).
• Once sliding occurs, the friction force drops to \(F = \mu_k N\), where the coefficient of kinetic friction \(\mu_k\) is usually 20 to 25 % lower than \(\mu_s\).
• To a good approximation, the friction force does not depend on the apparent contact area or the sliding speed.

These rules, often called Coulomb friction, are empirical approximations. Coefficients depend on the materials, surface finish, contamination, moisture and temperature, and vary widely; published values are guides, and safety-critical designs use conservative values or tests.

The resultant of N and the maximum friction force makes an angle φ with the normal, where \(\tan\phi_s = \mu_s\). This angle of friction gives a useful geometric test: a block on an incline is on the verge of sliding when the incline angle equals φ_s. For dry sand or gravel, the angle of repose, the steepest stable slope of a pile, is closely related.

Solving friction problems

Friction problems fall into three types. In the first, the loads are known and the question is whether the body stays put: solve for the friction force needed for equilibrium and compare it with μ_s N. In the second, motion is impending at every contact where slip could occur: set F = μ_s N there and solve for the unknown load. In the third, there are several possible ways for motion to start (slipping at one contact or another, or tipping instead of sliding), and each must be checked to see which happens first, at the smallest load.

Friction always acts to oppose the motion or impending motion of each surface relative to the other. Decide the direction of impending motion first, then draw the friction force on each body in the opposite direction. Lesson 4's ladder example is a classic case: the ladder can slip only at both ends at once, so friction must be at its limit at both contacts (or at the floor only, if the wall is smooth).

Wedges

A wedge is a simple machine that converts a small force pushing it in into a large force perpendicular to its faces. Wedges are used to lift heavy machinery and structures a few millimeters for leveling, to split materials, to shim and align equipment, and to hold things in place. Because the wedge slides on two or more surfaces at once, friction at every surface must be considered, and in most applications friction is large compared with the useful work, which is what makes a wedge stay put after it is driven: it is self-locking.

Screws

A screw thread is a wedge wrapped around a cylinder. Turning the screw slides the load up an incline whose slope is the lead angle θ, given by \(\tan\theta = \ell/(2\pi r)\), where ℓ is the lead (the distance the screw advances in one turn) and r is the mean radius of the thread. For a square thread, the moment needed to raise a load W is:

If θ < φ, the screw is self-locking: it holds its load without a brake, and a positive moment is needed to lower it. If θ > φ, the load drives the screw backward on its own, which is useful in some mechanisms but dangerous in a jack.

Belt and rope friction

When a flexible belt, rope or band wraps around a fixed drum, friction acting all around the contact allows a small force on one end to hold a much larger force on the other. Considering a small element of the belt in equilibrium and integrating around the angle of contact β gives the capstan equation:

where T₂ is the larger tension, on the side toward which slip would occur, T₁ the smaller, μ the coefficient of friction and β the wrap angle in radians. The exponential makes the effect powerful: each extra turn multiplies the holding capacity.

Belt drives use the same principle to transmit power: the difference between the tight-side and slack-side tensions, times the pulley radius, is the torque transmitted, and the belt slips if the tension ratio exceeds \(e^{\mu\beta}\). A flat belt with μ = 0.35 and 180° of wrap can sustain a tension ratio of 3.0. V-belts, wedged into a grooved pulley, behave as if μ were increased by a factor of 1/sin(groove half-angle), about 3 for a typical groove, which lets them transmit more power at lower tension. Band brakes on winches and some vehicles work the same way in reverse, using friction to absorb energy.

Friction in engineering practice

Friction appears in many design checks beyond the classic problems:

• Sliding of foundations and retaining walls. The base friction μN must exceed the horizontal load by a safety factor, typically 1.5, as in Lesson 7's wall, whose 192 kN/m weight on soil with μ = 0.5 resists 96 kN/m of sliding against a water force of 78.5 kN/m, a factor of only 1.22: the wall would need a key or more width.
• Slip-critical bolted joints. In some steel connections, high-strength bolts are tensioned so strongly that the clamped plates transfer load by friction rather than by bearing on the bolts, preventing slip under fatigue or reversing loads. The design slip resistance is the clamping force times a slip coefficient that depends on the surface preparation.
• Vehicle braking and traction. The maximum deceleration of a vehicle is about μg, so stopping distance from speed V is about \(V^2/(2\mu g)\): from 100 km/h, about 50 m on dry pavement (μ = 0.8) but over 300 m on ice (μ = 0.1), not counting reaction time. Highway designers use these values, with margins, to set stopping sight distances.
• Slips and falls. Floor slip resistance is measured and specified for walking surfaces, and falls on the same level are among the commonest workplace injuries, especially in winter.
• Bearings. Rolling-element bearings have effective friction coefficients around 0.001 to 0.005, a hundred times less than sliding, which is why they are used wherever shafts turn.