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?

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nuuk nuuk · Answer 1 · 2019-11-22T08:38:16+01:00

Answer:[tex]\theta =\tan ^{-1}(\frac{v^2}{gR})[/tex]

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

[tex] R\cos \theta [/tex] will balance weight and [tex]R\sin \theta [/tex] will provide the necessary centripetal Force

thus [tex]R\sin \theta =\frac{mv^2}{R}[/tex] ------------1

[tex]R\cos \theta =m g[/tex] ----------------2

Divide 1 & 2 we get

[tex]\frac{R\sin \theta }{R\cos \theta }=\frac{mv^2}{mgR}[/tex]

[tex]\tan \theta =\frac{v^2}{gR}[/tex]

[tex]\theta =\tan ^{-1}(\frac{v^2}{gR})[/tex]