How this instrument works
Tilt a curve and its normal force tilts with it. One component still carries weight; another now points at turn centre and becomes exactly the centripetal force circular motion demands. Set N·sin θ = m·v² ⁄ r beside N·cos θ = m·g, divide one by another, and you are left with tan θ = v² ⁄ (r·g). Mass vanishes in that division, which is why the loaded coach and the empty scooter want identical banking.
Christiaan Huygens had v² ⁄ r in hand by 1659, in a manuscript on centrifugal force that stayed unpublished until after his death, and he applied it to the conical pendulum — a bob swung in a horizontal circle, whose string hangs at precisely this angle. Newton folded that result into the Principia in 1687. Engineers met it again during the 1830s as railways began raising outer rails through curves, a practice still called cant in Britain and superelevation across North America.
Two assumptions do all of the work here. First, no friction: sideways grip is set to zero, so one design speed suits any given tilt, and anything quicker or slower needs rubber to cover the difference. Second, a point mass: track width, centre-of-gravity height, suspension roll and sidewall flex are absent, so nothing about rollover margin or how a real chassis leans shows up in an answer. Aircraft obey an identical relation with lift replacing normal force, though pilots also watch load factor, which grows as 1 ⁄ cos θ.
- Type your cornering speed into Speed — its unit menu accepts km/h, mph and knots as well as m/s.
- Enter Turn radius in metres, kilometres or feet, measured along a path your vehicle actually traces rather than along an outer kerb.
- Read Ideal bank angle (degrees): the tilt at which sideways friction falls to zero at that exact speed.
- Sweep Speed up and down to watch how quickly it climbs — doubling speed quadruples a tangent, though never quite an angle.
Worked example — 108 km/h through a 100 m curve
Picture a slip road of 100 m radius taken at 30 m/s, which is 108 km/h or 67 mph. Squaring speed gives 900 m²/s², while r·g comes to 100 × 9.80665 = 980.665 m²/s². Their ratio, 0.917745, is a tangent, so θ = arctan(0.917745) = 42.5439895162 degrees — printed here as 42.544°.
Picture that camber and it stops sounding routine: 42.5 degrees is close to a wall, steeper than any public highway on Earth. Highway codes cap superelevation near 12 per cent, roughly 6.8 degrees, with 6 per cent far more usual. At 6 per cent, an otherwise identical 100 m curve stays frictionless only up to 7.67 m/s, about 27.6 km/h. Every metre per second above that comes from tyre grip — which is exactly why designers widen the bend or post a lower limit instead of tilting tarmac until parked cars slide off it.
Questions
Does a heavier vehicle need a steeper bank?
No. Mass sits on both sides of that force balance and cancels during division, so the 40-tonne truck and the bicycle want one angle at matching speed and radius. Weight still governs tyre loading, rollover threshold and stopping distance — none of which this relation describes. For angle alone, mass drops out completely.
What happens above or below the design speed?
In a frictionless idealisation, going quicker slides you up and out of a bend, going slower slides you down into it. Real surfaces supply grip, so banking serves a band of speeds around its design value rather than one exact number. That band widens with better rubber and narrows sharply on ice, where a frictionless answer becomes uncomfortably honest.
Which units belong in this formula?
Speed in metres per second, radius in metres, gravity in m/s². The ratio v²/(r·g) is dimensionless, and its arctangent means nothing unless units agree. Menus on Speed and Turn radius convert for you. Pushing raw km/h through a bare formula inflates v² by 12.96 and drives almost any result toward 90 degrees — much the commonest mistake with this quantity.
Is a motorcycle's lean angle really the same calculation?
Yes, with one shift of reference. On flat ground a rider leans until a line from contact patch to centre of mass tilts from vertical by arctan(v²/(r·g)) — an identical expression, only measured from vertical instead of from horizontal. Cornering at 30 m/s on a 100 m radius therefore means leaning 42.5 degrees, near where a sports bike starts grinding hard parts.
Why do velodromes and speedway ovals bank so much steeper than roads?
Because nothing slow ever uses them. Daytona's turns reach 31 degrees and track cycling banks past 40, which works only while every machine on that surface stays fast. A public road must also hold a stopped delivery van in freezing rain, so codes keep camber gentle and let tyres cover what remains.
How does bank angle relate to load factor in an aircraft?
Load factor equals 1 ⁄ cos θ. Banking 42.544 degrees pulls about 1.36 g; 60 degrees pulls exactly 2 g; 75 degrees nearly 3.9. Because lift must rise to hold altitude while turning, steep banks raise stall speed by a factor of √n — one reason steep-turn practice happens with height to spare.