Question

A curve of radius 20 m is banked so that a 1100 kg car traveling at 30 km/h can round it even if the road is so icy that the coefficient of static friction is approximately zero. The acceleration of gravity is 9.81 m/s 2 .

Find the minimum speed at which a car can travel around this curve without skidding if the coefficient of static friction between the road and the tires is 0.3. Answer in units of m/s.

Answer 1

the frictional force is not µmgcosΘ, because the normal force has increased to mgcosφ where φ is the angle between the resultant force on the car Fr and the normal to the track surface. Fr itself is m√((V²/r)²+g²) and its angle from the vertical ß = arctan(V²/rg).

At 30 km/h = 8.333m/s,

radius 20 m

Letting the bank angle of the track be Θ,
φ = ß-Θ and putting the rest of it together,
µ = tanφ
ß = φ+Θ and
V² = rg*tanß
Using your numbers,
Θ = arctan(8.333²/(20*9.81)) = 19.489°
φ = arctan(0.3) = 16.699° →
ß = (Θ + φ ) = 36.188° and finally,

V = √(rgtanß)

= √(20*9.81*tan36.188) = 11.980 m/s = 43.128 km/hr