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Question
A racing car travels on a track (without banking) ABCDEFA (Figure). ABC is a circular arc of radius 2 R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is µ = 0.1. The maximum speed of the car is 50 ms–1. Find the minimum time for completing one round.
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Solution
Balancing frictional force for centripetal force = `(mv^2)/r` = f = μN = μmg
Where N is the normal reaction
∴ `v = sqrt(μrg)` ....(Where, r is radius of the circular track)
For path ABC, Path length = `3/4 (2π 2R) = 3π R = 3π xx 100`
= 300 πm
`v_1 = sqrt(μ2Rg) = sqrt(0.1 xx 2 xx 100 xx 10)`
= 14.14 m/s
∴ `t_1 = (300π)/(14.14)` = 66.6 s
For path DEF, Path length = `1/4 (2πR) = (π xx 100)/2` = 50π
`v_2 = sqrt(μRg) = sqrt(0.1 xx 100 xx 10)` = 10 m/s
`t_2 = (50π)/10` = 5π s = 15.7 s
For path, CD and FA
Path length = R + R = 2R = 200 m
`t_3 = 200/50` = 4.0 s
∴ Total time for completing one round t = t1 + t2 + t3 = 66.6 + 15.7 + 4.0 = 86.3 s
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