physics problem

Question

A car moving at speed v enters a curve of radius R. It is possible to bank the curve at just the right angle so that you don’t need friction to maintain the car’s turning radius. Your engineering firm plans to rebuild the curve so that a car moving at a chosen speed can safely make the turn in the absence of friction. At what angle b (for a given v) should the curve be banked?

Answer 1

In the absence of friction (imagine an icy road as an approximate) there are only 2 forces acting on the car

first gravitational Force = mg in vertical direction downward toward the center of the earth

second N the normal force from the slope (banked road)

if you draw your free body diagram you notice that N has an angle of θ from the vertical toward the center of the curve.

if you decompose N in to its horizontal and Vertical Component you should see:

N cosθ upward in vertical direction

N sinθ toward the center of the curve

if you want to have a stable trajectory in the road. you need the net force be equal to the centripetal force

and the vertical forces should cancel each other, that is:

mg = N cosθ

Fcen= N sinθ = mv²/R

replacing N from the vertical equilibrium equation => (mg/cosθ)sinθ=mv²/R => tanθ = v²/Rθg

or θ=tan^-1(v²/Rg)