A racing car travels on a track (without banking) `ABCDEFA`.`ABC` is a circular arc of radius `2R. CD` and `FA` are straight paths of length `R` and `DEF` is a circular arc of radius `R = 100m`. The co-efficient of friction on the road is `¼ =0.1`. the maximum speed of the car is `50m s-1`. Find the minimum time for completing one round.
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Explanation

As the track is unbanked the necessary centripetal force is provided by force of friction
`(m upsilon^(2))/(r) = F = mu R = mu mg :. upsilon= sqrt(mu r g)`
For path ABC : length `= (3)/(4) (2pi2R) = 3 pi R = 3 pi xx 100 = 300 pi m`
`upsilon_(1)=sqrt(mu 2 Rg) = sqrt( 0.1 xx 2 xx 100 xx 10) = 14.14 m//s`
`:. t_(1) = (300 pi)/(14.14) = 66.6s`
For path DEF: length `= (1)/(4) (2pi R) = (pi xx 100)/(2) = 50pi`
`upsilon_(2)=sqrt(muRg) = sqrt(0.1 xx 100 xx 10 )= 10 m//s`
`t_(2) = (50pi)/(10) = 5 pi sec = 15.7 s`
For paths `CD` and `FA`
length `=R + R =2 R = 200 m`
`t_(3) = (200)/(50) = 4.0 s`
`:.` Total time for completing one round `t = t_(1) + t_(2) + t_(3) = 66.6 + 15.7 + 4.0 = 86.3 s`.