Question

A civil engineer plans to design a curved ramp such that a car may not have to rely on friction to round the curve without skidding. The roadway is tilted toward the inside of the curve. Suppose the designated speed is v and the radius of the curve is R. At what angle should the curve be tilted?

Answer 1

Answer:$\theta =\tan ^{-1}(\frac{v^2}{gR})$

Explanation:

let \theta be the inclination at which curve is tilted

v is the speed of car and R is the radius of curve

m is the mass of car

Suppose R is the reaction offered by road to car

Resolving R in x and y direction we get

$R\cos \theta$ will balance weight and $R\sin \theta$ will provide the necessary centripetal Force

thus $R\sin \theta =\frac{mv^2}{R}$  ------------1

$R\cos \theta =m g$  ----------------2

Divide 1 & 2 we get

$\frac{R\sin \theta }{R\cos \theta }=\frac{mv^2}{mgR}$

$\tan \theta =\frac{v^2}{gR}$

$\theta =\tan ^{-1}(\frac{v^2}{gR})$