For a vehicle moving on a banked curved road, using free body diagram (FBD), obtain the formula for the maximum safe speed `v_(max)`.

Explanation

(a) A road banked through angle `theta` and having radius R is shown .

Three forcess act on it .
(1) Weight of the vehicle (2) Normal reaction force (3) frictional force acting away the road surface
Taking the components of normal reaction force
(1) N cos `theta ` vertical direction
N s in `theta` along horizontal direction
these five act on the vehicle as shown in the figure
from the figure `n cos theta = mg + f sin theta`
`:. mg = N cos theta - f sin theta .... (1)`
the components of normal force N and frictional force f provide necessary centripetal force
`(v^(2))/(Rg) = (n sin theta)/( n cos theta) `
Frictional force `f_(m)` developed due to maximum permissible speed is given by
`(f_(s))_(max) = mu_(s) N`
In this equation the massss of the vehicle and mass of passenger are not included it means that permissible speed does not depend on the mass of the vehicle or passenges
the above equation shows that maximum permissible speed of a vehicle is more than on a banked circular road than on plain circular road.