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 ( g R ^(2) (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) ( mu _(s) + tan theta)/( 1 - mu _(s ) tan theta))`
D. `sqrt ((g)/( R ^(2)) ( mu _(s) + tan theta)/( 1 - mu _(s) tan theta))`

Answer

B

Explanation

`N = mg cos theta + ( mv ^(2))/( R) sin theta`
`f _(max) = mu N implies `
`f _(max) = mu _(s) mg cos theta + ( mu _(s) mv ^(2))/( R) sin theta`
`mg sin theta + f _( max) = ( mv ^(2))/(R) cos theta`
`mg sin theta + mu _(s) mg cos theta + ( mu _(s) mv ^(2))/( R) sin theta`
`= ( mv ^(2))/( R) cos theta`
`g sin theta + mu _(s) g cos theta`
`= ( v ^(2))/(R) (cos theta - mu _(s) sin theta)`
`g R [ ( tan theta + mu _(s))/( 1 - mu _(s) tan theta) ] = v ^(2)`