For maximum velocity, f = μsN( i) becomes:
Ncosϴ = μsNsinϴ + mg Or, Ncosϴ- μsNsinϴ = mg Or, N = mg /( cosϴ- μssinϴ)
Put the above value of N in( ii) μsNcosϴ + Nsinϴ = mv 2 / r μsmgcosϴ /( cosϴ- μssinϴ) + mgsinϴ /( cosϴ- μssinϴ) = mv 2 / r mg( sinϴ + μscosϴ)/( cosϴ- μssinϴ) = mv 2 / r Divide the Numerator & Denominator by cosϴ, we get v 2 = Rg( tanϴ + μs) /( 1- μs tanϴ) v = √ Rg( tanϴ + μs) /( 1- μs tanϴ) This is the miximum speed of a car on a banked road.
Special case: When the velocity of the car = v0, o No f is needed to provide the centripetal force.( μs = 0) o Little wear & tear of tyres take place. vo = √ Rg( tanϴ)
Problem: A circular racetrack of radius 300 m is banked at an angle of 15 °. If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the
( a) optimum speed of the racecar to avoid wear and tear on its tyres, and( b) maximum permissible speed to avoid slipping?