A car is negotiating a curved road of radius R. The road is banked at angle `theta`. The coefficeint 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((mu_(s)+tan theta)/(1-mu_(s) tan theta)))`
B. `sqrt((g)/(R)((mu_(s)+tan theta)/(1-mu_(s) tan theta)))`
C. `sqrt((g)/(R)^(2)((mu_(s)+tan theta)/(1-mu_(s) tan theta)))`
D. `sqrt(gR^(2)((mu_(s)+tan theta)/(1-mu_(s) tan theta)))`

Answer

A

Explanation

According to question, a car is negotiating a curved road of radius R. The road is banked at angle `theta` and the coefficient of friction between the tyres of car and the road is `mu_(s)`. So, this given situation can be drawn as shown as in figure below.

Considering the case of vertical equilibrium
`N cos theta=mg+f_(l) sin theta`
`rArr" "mg=N cos theta-f_(l) sin theta` ....(i)
Considering the case of horizontal equilibrium,
`Nsin theta+f_(l)cos theta=(mv^(2))/(R)`....(ii)
On dividing Eq. (i) by Eq. (ii), we get
`(V^(2))/(Rg)=(sintheta+mu_(s)costheta)/(costheta-mu_(s)sintheta)" "[f_(l)propmu_(s)]`
`rArrv=sqrt(Rg((sintheta+mu_(s)costheta)/(costheta-mu_(s)sintheta)))rArrv=sqrt(Rg((tantheta+mu_(s))/(1-mu_(s)tantheta)))`