Static and kinetic friction
Friction is the force that opposes relative motion between two surfaces in contact. It acts along the surface, opposite to the motion (or to the tendency of motion). It comes from tiny bumps on surfaces locking into each other and from molecular attraction at the contact points.
Static friction fs
It acts when a body is pushed but does not move. It is self-adjusting: push with 5 N, it is 5 N; push with 10 N, it is 10 N. It has a maximum value called limiting friction:
fs ≤ fs,max = μs N
μs is the coefficient of static friction and N is the normal force.
Kinetic friction fk
Once sliding begins, fk = μk N. Usually μk < μs, so it is harder to start moving a heavy box than to keep it moving.
Graph of friction against applied force
Friction rises along a 45° line (equal to the pull), reaches the limiting value, then drops a little to the kinetic value and stays flat.
Angle of repose
Tilt a board until a block just begins to slide. At that angle θ, mg sin θ = μs mg cos θ, so μs = tan θ.
Laws of friction
- Limiting friction is proportional to the normal force: fs,max = μs N.
- It does not depend on the area of contact (for the same N and surfaces).
- It depends on the nature and condition of the two surfaces (wood, rubber, wet, polished), through μ.
- Kinetic friction is also proportional to N, is a little less than limiting friction, and hardly depends on speed (for ordinary speeds).
μ has no unit. Typical values: wood on wood ≈ 0.4–0.5, rubber on dry road ≈ 0.7–0.9, steel on ice ≈ 0.02–0.1.
Friction on an inclined surface: here N = mg cos θ, not mg. On a flat floor with a pull at an angle upward, N = mg − P sin θ.
Friction is needed (walking, brakes, writing, holding things) but also wastes energy as heat and wears out parts. So it is a "necessary evil".
Rolling friction
When a wheel or ball rolls without slipping, the contact point is momentarily at rest. The small force that still opposes rolling is rolling friction. It comes mostly from the surfaces getting slightly pressed out of shape.
Rolling friction is much smaller than sliding friction (often 100 times smaller). That is why the wheel was such a big invention, and why suitcases have wheels.
Ball bearings and roller bearings between an axle and a hub change sliding into rolling and cut friction greatly.
Lubrication and other ways to reduce friction
A lubricant (oil, grease, graphite powder) forms a thin layer between two surfaces. The surfaces no longer rub directly; the layers of lubricant slide over each other easily. This reduces friction, heat and wear.
- Oiling cycle chains and hinges; engine oil in vehicles.
- Polishing surfaces (but very smooth surfaces can stick again due to molecular attraction).
- Using ball bearings.
- Streamlined shapes to reduce air or water drag.
- A cushion of compressed air (hovercraft).
To increase friction we use treads on tyres, sand on icy roads, and chalk powder on a gymnast's hands.
Try it at home
Book on a ramp: put a coin on a hardcover book and lift one end slowly. Measure the height h and length L when the coin just slides. μs = tan θ = h/√(L² − h²).
Area test: pull a brick-shaped soap box with a rubber band, once lying flat, once on its side. The band stretches about the same amount.
Rolling test: push a book across a table, then put it on 4 round pencils and push again. Much easier!
Key formulas and definitions
- fs ≤ μs N (static, self-adjusting)
- fs,max = μs N (limiting friction)
- fk = μk N, μk < μs
- μs = tan θ (angle of repose)
- On incline: N = mg cos θ
- Net force on sliding block: P − μk N = m a
Worked examples
1. A 5 kg block is on a floor with μs = 0.5. Find the limiting friction (g = 10 m/s²).
N = mg = 50 N. fs,max = 0.5 × 50 = 25 N.
2. The same block is pulled with 15 N. What is the friction force?
15 N < 25 N, so the block stays at rest and static friction = 15 N (only as much as needed).
3. The block (μk = 0.4) is pulled with 30 N. Find its acceleration.
fk = 0.4 × 50 = 20 N. Net force = 30 − 20 = 10 N. a = 10/5 = 2 m/s².
4. A car at 20 m/s brakes and skids to a stop. μk = 0.5. Find the stopping distance (g = 10 m/s²).
Deceleration a = μk g = 5 m/s². v² = u² − 2as → 0 = 400 − 10 s → s = 40 m.
5. A block just starts to slide when a plank is tilted to 30°. Find μs.
μs = tan 30° = 1/√3 ≈ 0.58.
6. A 10 kg box on a floor (μk = 0.3) is pulled by a rope at 37° above the horizontal with 50 N. Find the acceleration (sin 37° = 0.6, cos 37° = 0.8, g = 10).
N = mg − P sin 37° = 100 − 30 = 70 N. fk = 0.3 × 70 = 21 N. Horizontal pull = 50 × 0.8 = 40 N. a = (40 − 21)/10 = 1.9 m/s².
7. A 2 kg block slides down a 30° incline with μk = 0.2. Find its acceleration (g = 10).
Down the slope: mg sin 30° = 10 N. N = mg cos 30° = 17.32 N, fk = 0.2 × 17.32 = 3.46 N. a = (10 − 3.46)/2 ≈ 3.27 m/s².
Common mistakes
- Always writing friction = μN. For a body at rest, static friction equals only the force needed to hold it, which may be less than μs N.
- Taking N = mg on an incline or when a pull has an upward part. Find N from the forces perpendicular to the surface.
- Thinking a larger contact area means more friction. For the same N, area does not matter.
- Thinking friction always opposes motion of the body. It opposes relative motion of surfaces; for walking, friction on the foot points forward.