Any physicists? Please help.?

Question

An engineer wishes to design a curved exit ramp for a toll road in such a way that a car will not have to rely on friction to round the curve without skidding. He does so by banking the road in such a way that the necessary force causeinf the centripetal acceleration will be supplied by the component of the normal force toward the centere of the circular path. A) Show that for a given speed of v and a radius of r, the curve must be banked at the angle theta such that tan theta is v^2/ rg. B) Find the angle at which the curve should be banked if a typical car rounds it at a 50 m. radius and a speed of 13.4 m/s

Answer 1

If the road were not banked, one has to rely only upon the frictional force which provides the necessary centripetal force to turn around a curve.

If the road is banked, that is tilted to an angle C (called the angle of friction) the frictional force will disappear.

But the horizontal component of normal reaction will now provide the necessary centripetal force.

N sin C = M V V / R.

The vertical component of normal reaction is equal but opposite to the weight of the car.

N cos C = Mg.

Dividing the first by second equation,

Tan C = V V / (R*g) from which C can be calculated.

B).

Tan C = 13.4 x 13.4 / 50 x9.8

------------- = 0.3664.

C = 20.1 degree.

Answer 2

Draw free body diagram, resolve components, solve for theta.