Centripetal forces with cars

Question

A car can round a banked curve at an 84 m radius and a maximum speed of 16 m/s without relying on friction. What is the angle does the car bank when it turns?

 

Answer 1

Assume the road is banked at angle θ above the horizontal.

F_c = mv²/r

N = normal force exerted by surface of road on car

N sinθ = component of normal force in the radial direction.  This is the same as F_c.

 

N cosθ = component of normal force in vertical direction.  This must balance the weight of the car.

 

N cosθ = mg ⇒

 

N = mg/cosθ

 

mv²/r = N sinθ = (mg/cosθ)sinθ = mg tanθ

 

v²/r = g tanθ

 

θ = arctan[v²/(rg)]

 

v²/(rg) = 16²/[84(9.8)] ≅ 0.311

 

θ = arctan(0.311) ≅ 17.3º