How Fast Can a Banked Curve Be Taken Before and After Rain?
A new two-lane highway curves with a radius of 120 m and is banked at 10 degrees to the horizontal. The posted advisory speed is 80 km/h. During the dry season the coefficient of static friction between tyres and asphalt is about 0.70; after the first monsoon showers it falls to about 0.35, and after a diesel spill it can drop to 0.05. The design engineer argues that no speed restriction is needed because the bank was 'designed for the curve', while a traffic officer insists the 80 km/h limit must be lowered in the wet. A loaded truck of mass 12 000 kg and a small car of mass 900 kg use the curve. Take g = 9.8 m s^-2 and tan 10 degrees = 0.176, and remember that at very low speeds a vehicle may tend to slide down the incline.
1. Derive or state the expression for the maximum safe speed on a banked road with friction, and evaluate it for the dry and the wet road.
Read the worked answer
Resolving along and perpendicular to the incline for the limiting case in which friction acts down the slope:
- perpendicular: N cos(theta) = mg + f sin(theta)
- along the slope: N sin(theta) + f cos(theta) = mv^2/r
Setting f = mu N and eliminating N gives
v_max = sqrt[ rg (mu + tan(theta)) / (1 - mu tan(theta)) ].
Dry, mu = 0.70: (mu + tan theta)/(1 - mu tan theta) = (0.70 + 0.176)/(1 - 0.70 x 0.176) = 0.876/0.877 = 0.9997. v_max = sqrt(120 x 9.8 x 0.9997) = sqrt(1176) = 34.3 m s^-1 = 123 km/h.
Wet, mu = 0.35: (0.35 + 0.176)/(1 - 0.35 x 0.176) = 0.526/0.938 = 0.561. v_max = sqrt(1176 x 0.561) = sqrt(660) = 25.7 m s^-1 = 92 km/h.
Note that the answer does not contain m: both the centripetal force needed and the friction available scale with mass.
2. Is the posted 80 km/h limit safe in the wet? What happens on the diesel spill? Also find the minimum speed below which a vehicle would slide down the bank in each case.
Read the worked answer
80 km/h = 80/3.6 = 22.2 m s^-1.
Wet road: v_max = 25.7 m s^-1, so 22.2 m s^-1 is inside the limit but with only a 14 % margin, and that margin is consumed by braking, worn tyres or a driver taking a slightly wider line. The officer is right to want the limit reviewed; 80 km/h is defensible only as an advisory, not as a guarantee.
Diesel spill, mu = 0.05: (0.176 + 0.05)/(1 - 0.05 x 0.176) = 0.226/0.991 = 0.228. v_max = sqrt(1176 x 0.228) = sqrt(268) = 16.4 m s^-1 = 59 km/h. The posted 80 km/h is now physically impossible to hold; the road must be closed or the spill treated.
Minimum speed, v_min = sqrt[ rg (tan theta - mu)/(1 + mu tan theta) ]:
- dry and wet: tan theta - mu is negative (0.176 < 0.35), so the expression has no real value: friction alone can hold a stationary vehicle on the incline and there is no lower limit.
- diesel spill: (0.176 - 0.05)/(1 + 0.0088) = 0.125; v_min = sqrt(1176 x 0.125) = 12.1 m s^-1 = 44 km/h. On the slippery surface a slow-moving vehicle is now the one in danger of sliding down the bank, which is the opposite of the usual intuition.
3. Evaluate the engineer's claim that 'the bank was designed for the curve'. What is the maximum speed the banking alone can support, and why does the mass-independence of the formula not make the heavy truck as safe as the car?
Read the worked answer
With no friction at all (mu = 0) the formula reduces to v = sqrt(rg tan theta) = sqrt(120 x 9.8 x 0.176) = sqrt(207) = 14.4 m s^-1 = 52 km/h. So the banking alone supports only about 52 km/h, far below the posted 80 km/h; the engineer's claim is false. The bank reduces the reliance on friction but does not remove it.
The vanishing of m from v_max = sqrt[rg(mu + tan theta)/(1 - mu tan theta)] is a genuine result of the point-mass model: the required centripetal force and the maximum available friction are both proportional to m, so their ratio is fixed. But the model is only as good as its assumptions. A loaded truck has (i) a much higher centre of mass, so the normal reaction is redistributed and the vehicle can roll before it slips, (ii) a larger contact patch with different rubber and different mu, and (iii) a longer wheelbase that makes it track a wider arc than the nominal radius. For these reasons the mass-independent formula gives an optimistic upper bound for the truck, and the safe operating speed must be set from tests on the actual vehicle.