A car is moving along a banked road laid out as a circle of radius r . (a) What should be the banking angle `theta` so that the car travelling at speed v needs no frictional force from tyres to negotiate the turn ? (b) The coefficient of friction between tyres and road are `mu_(s) = 0.9` and `mu_(k) = 0.8` . At what maximum speed can a car enter the curve without sliding toward the top edge of the banked turn ?

Explanation

`N sin theta=(mv^(2))/(r)and N cos theta=mg Rightarrow tan theta=(v^(2))/(rg)`
NOTE: In above case friction does not play any role in negotiating the turn.
(b) If the driver moves faster than the speed mentioned above, a friction force myst act parallel to the road, inward towards center of the turn.
`Rightarrow Fcostheta+Nsintheta=(mv^(2))/(r)and N cos theta=mg+f sin theta`
For maximum speed of f=`mu N`
`N (mucos theta+sin theta)=(mv^(2))/(r)and N (cos theta-musin theta)=mg`
`=(v^(2))/(rg)=(sin theta+mucostheta)/(cos theta-mu sin theta)Rightarrow v=sqrt(((sin theta+mu cos theta)/(cos theta-mu sin thea)))rg`