question_answer 77)

                  A racing car travels on a track (without banking) ABCDEFA (Fig.) 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 = 100 m. The co-efficient of friction on the road is \[\mu =\,0.1\]. The maximum speed of the car is \[50\,m\,{{s}^{-1}}\]. Find the minimum time for completing one round.

Answer:

                  A racing can negotiate a circular track (without banking) if \[\frac{m{{\upsilon }^{2}}}{r}=\,\mu \,mg\]                 or \[\upsilon =\,\sqrt{\mu gr}\]                 For circular track ABC, \[\upsilon =\,\sqrt{0.1\,\times \,10\times \,200}\]                 \[=14.14\,\,m{{s}^{1}}\]                 For circular track DEF, \[\upsilon =\,\sqrt{0.1\,\times \,10\times 100}\]                 \[=10\text{ }m{{s}^{1}}\]                 Here, speed on a straight path, \[{{\upsilon }_{1}}=50\,m{{s}^{-1}}\] Time for completing one round = Time to complete ABC + time to complete CD + time to complete DEF + time to complete                 \[FA\,=\,\frac{\frac{3}{2}\,\times \,2\pi \times \,2R}{14.14}\,+\frac{R}{50}+\,\frac{\frac{1}{4}\,\times 2\pi R}{10}+\,\frac{R}{50}\]                 = 97.4s.