Q. 37 A racing car travels on a track (without banking) ABCDEFA. 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 coefficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 m/s. Find the minimum time for completing one round.

Hint: Balance frictional force with centripetal force for the circular path.
Step 1: Find maximum speed of the car on the circular path.
Balancing frictional force with centripetal force
mv2r=f=μN=μmg
 
where N is normal reaction.
v=μrg (where, r is radius of the circular track) 
Step 2: Find the time taken for path ABC.
For path ABC, path length =34(2π2R)=3πR=3π×100
=300πmv1=μ2Rg=0.1×2×100×10=14.14m/st1=300π14.14=66.6s
Step 3: Find the time taken for path DEF.
ForpathDEF,pathlength=14(2πR)=π×1002=50π
v2=μAg=0.1×100×10=10m/s
t2=50π10=5πs=15.7s
Step4:FindtimetakenforpathCDandFA.
Forpaths, CD and FA,pathlength=R+R=2R=200m
t3=20050=4.0s
Step5:Findtotaltimetakenforcompletingoneround.
 Total time for completing one round t=t1+t2+t3=66.6+15.7+4.0=86.3s.