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

Balancing frictional force for centripetal force `(mv^(2))/(r) = f = mu N = mu mg`
where N is normal reaction
`therefore " " v = sqrt(murg) " "` (where r is radius of the circular track)
For path ABC `" "` Path length = `(3)/(4)(2pi 2R) = 3pi R = 3 pi xx 100`
=`300 pi m`
`v_(1) = sqrt(mu2Rg) = sqrt(0.1 xx 2 xx 100 xx 10)`
= `14.14` m/s
`therefore " " t_(1) = (300pi)/(14.14) = 66.6 s`
For path DEF Path length = `(1)/(4) (2 pi R) = (pi xx 100)/(2) = 50 pi`
`v_(2) = sqrt(muRg) = sqrt(0.1 xx 100 xx 10) = 10` m/s
`t_(2) = (50 pi)/( 10) = 5 pi s = 15.7 s`
For paths CD and FA
Path length = R + R = 2R = 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 `