Banked track: from the survey stakes to the cornering force
SPH4U Grade 12 Physics · Dynamics
A closed test oval is being commissioned. On the banked north turn a surveyor levels across one 3.50 m lane and finds the outer edge of the pavement sitting 1400 mm above the inner edge; the turn itself is laid out on a 175 m radius. Find the bank angle, the speed a car could hold through the turn with no help at all from sideways friction, the centripetal acceleration at that speed, and the inward force the pavement must supply to a 1520 kg car.
The bank angle is pure geometry — rise over horizontal run — and it has to come first, because every dynamics question about this turn is really a question about this one angle. The rise comes off the level in millimetres and the lane width in metres, so watch the conversion: 1400 mm across 3.50 m is a ratio of 0.4, not 400.
Carried onward at full precision, not this rounded figure.
Now the physics. At the design speed the tilted normal force alone supplies mv²/r, so tan θ = v²/rg and the angle from step 1 hands over a speed. Note the mass never enters — the same tilt works for a motorbike.
Carried onward at full precision, not this rounded figure.
Feed the design speed back into a = v²/r. The result is g tan θ, which is a useful thing to recognise on sight.
Carried onward at full precision, not this rounded figure.
Newton's second law converts that acceleration into the force the road has to deliver. This is the number a structural engineer needs; the driver only feels it.
Carried onward at full precision, not this rounded figure.
Why this order
The order here is the whole lesson. A bank angle is a geometric fact about a piece of concrete, measurable with a level and a tape before a single car drives on it, so it goes first. Only then does dynamics get a say: tan θ = v²/rg turns that fixed geometry into one particular speed, the design speed, at which the normal force does all the work and friction does none. Below it a car tends to slide down the bank; above it, friction has to make up the difference, and when the tyres run out the car goes over the top. A single bank angle is not a safety margin — it is a single speed, with tolerance on either side paid for in grip.
Two things trip students between step 1 and step 2. The first is units, and it bites twice. The level reading arrives in millimetres and the lane width in metres, so a ratio taken off the raw numbers is 400 instead of 0.4 and the bank comes out at 89.86°; then the answer to step 1 is an angle, which this site carries in radians even though it shows you degrees, and a calculator left in the wrong mode produces a design speed that is wrong by nothing recognisable. The second is the disappearing mass. Students expect a heavy car to need a steeper bank, but writing N sin θ = mv²/r beside N cos θ = mg and dividing kills the m outright — which is why step 2 needs no mass at all and step 4 suddenly does. Mass decides how much force, never how much tilt. Railway engineers hit the same result in the 1830s and still call it cant; Daytona's 31° and Talladega's 33° are this equation built at the limit of what a paving machine can hold.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.