Friction or the Normal Force Supplies the Centripetal Force: Write That Balance
A vehicle going round a bend needs a net inward force of m v²/R. On a flat road only friction can supply it; on a banked road the normal force can supply it by itself at one speed, with friction helping beyond it.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Flat road and rotating discs
A coin on a record turning at ω stays put if m r ω² ≤ μ m g, so r ≤ μ g/ω². A motorcyclist leaning to avoid skidding on a flat road leans at tan θ = v²/(R g) from the vertical: at 72 km/h (20 m/s) on a bend of radius 20 m, tan θ = 400/200 = 2.
Banked roads
At a fixed angle, v² is proportional to R, so to raise the safe speed by 10% the radius must be multiplied by 1.1² = 1.21: from 20 m to 24.2 m. Friction extends the range: v_max² = R g (tan θ + μ)/(1 - μ tan θ) and v_min² = R g (tan θ - μ)/(1 + μ tan θ).
OlympIQ exam lens
- ✓Flat road: v_max = sqrt(μs R g); coin on a disc: r ≤ μ g/ω²
- ✓Banked road without friction: tan θ = v²/(R g)
- ✓At a fixed banking angle, v² is proportional to R
- ✓The lean angle of a cyclist from the vertical: tan θ = v²/(R g)
Do not convert the speed incorrectly. Kilometres per hour must be divided by 3.6 before using v²/(R g), so 72 km/h is 20 m/s, not 72 m/s.
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