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.
.

Options

A. `50 s`
B. `90.3 s`
C. `83.6 s`
D. `86.3 s`

Answer

D

Explanation

Balancing frictional force for centripetal
force `(mv^(2))/(r ) = f = mu N =mumg`
where, N is normal reaction
`:. v =sqrt(murg)` (where r is radius of the circular track)
For path `ABC` Path length
` =(3)/(4) (2pi 2R) =3pi R =3pi xx 100`
`= 300 pi`
`v_(1) =sqrt(mu2Rg)=sqrt(0.1 xx 2 xx 100 xx 10)`
`=14 .14 m//s`
`t_(1) = (300pi)/(14.14) =66.6s`
For path `DEF` Path
length `=(1)/(4) (2piR) = (pi xx 100)/(2) =50pi`
`v_(2) = sqrt(muRg) = sqrt(0.1 xx 100 xx 10) =10m//s`
`t_(2) = (50pi)/(10) =5pis =15.7s`
For paths, `CD` and `FA`
Path length
`= R +R =2R =200m`
`t_(3) = (200)/(50) =4.0s`
`:.` Total time for completing one round
`t =t_(1) +t_(2) +t_(3) =66.6 + 15.7 + 4.0 =86.3s` .