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 ?