A racing car travels on a track (without banking) ABCDEFA (Fig.). ABC

1.	A racing car travels on a track (without banking) ABCDEFA (Fig.). 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 m = 0.1. the maximum speed of the car is 50 ms-1. Find the minimum time for completing one round.
Answer» The centripetal force to keep car in circular motion is provided by frictional force (inward to centre). For DEF \(m\frac{v^2}{R}\) = mgμ v_max = \(\sqrt{gμR}\) = \(\sqrt{100}\) = 10 ms^-1 For ABC \(\frac{v^2}{R}\) = gμ, v = \(\sqrt{200}\) = 14.14 ms^-1 Time for DEF = \(\frac{π}{2}\times \frac{100}{10}\) = 5 πs Time for ABC = \(\frac{3π}{2} \frac{200}{14.14}\) = \(\frac{300π}{14.14}s\) For FA and DC = 2 × \(\frac{100}{50}\) = 4s Total time = 5π + \(\frac{300π}{14.14}\) + 4 = 86.3s

A racing car travels on a track (without banking) ABCDEFA (Fig.). 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 m = 0.1. the maximum speed of the car is 50 ms-1. Find the minimum time for completing one round.

Answer»

The centripetal force to keep car in circular motion is provided by frictional force (inward to centre).

For DEF

\(m\frac{v^2}{R}\) = mgμ

v_max = \(\sqrt{gμR}\) = \(\sqrt{100}\) = 10 ms^-1

For ABC

\(\frac{v^2}{R}\) = gμ, v = \(\sqrt{200}\) = 14.14 ms^-1

Time for DEF = \(\frac{π}{2}\times \frac{100}{10}\) = 5 πs

Time for ABC = \(\frac{3π}{2} \frac{200}{14.14}\) = \(\frac{300π}{14.14}s\)

For FA and DC = 2 × \(\frac{100}{50}\) = 4s

Total time = 5π + \(\frac{300π}{14.14}\) + 4 = 86.3s