A car is negotisting a curved road of radius `R` . The road is banked at an angle theta. The coefficient of friction between the tyres of the car and the road is `mu_(s)` . The maximum safe velocity on this road is:

Options

A. `sqrt(gR^(2)(mu_(s)+tantheta)/(1-mu_(s)tantheta))`
B. `sqrt(gR(mu_(s)+tantheta)/(1-mu_(s)tantheta))`
C. `sqrt(g/R(mu_(s)+tantheta)/(1-mu_(s)tantheta))`
D. `sqrt(g/R^(2)(mu_(s)+tantheta)/(1-mu_(s)tantheta))`

Answer

B

Explanation

Vertical equilibrium
`Ncoshteta=mg+f_(L)sintheta`
implies `mg=Ncostheta-f_(L)sintheta`
(i) Horizontal equilibrium
(ii) `Nsintheta+f_(L)costheta=(mv^(2))/(R)`
`(Eqn.(2))/(Eqn.(1))v^(2)/(Rg)=(sintheta+mu_(s)costheta)/(costheta+mu_(s)sintheta)`
implies `v=sqrt(Rg((sintheta+mu_(s)costheta)/(costheta+mu_(s)sintheta)))`
`sqrt(Rg((tantheta+mu_(s))/(1-+mu_(s)tantheta)))`