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) + tan theta)/(1-mu_(s)tan theta))`
B. `sqrt(gR(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") = muN rArr f_("max") = mu_(s)mg cos theta + (mu_(s)mv^(2))/R sin theta`
`mg sin theta + f_("max") = (mv^(2))/R cos theta`
`g sin theta + mu_(s)g cos theta = v^(2)/R (cos theta - mu_(s) sin theta)`
`gR[(tan theta + mu_(s))/(1 - mu_(s) tan theta)] = v^(2)`