Question

Find the minimum velocity of a vehicle of mass m on a banked road of banking angle θ, if coefficient of static friction of the wheels of vehicle with the road is µ?

Answer 1

Dear student

$$\begin{gathered} Thefreebodydiagramofthecarmovingonaninclinedplaneisasshowninthefigure.Inthefreebodydiagramfrictonactsuptheplaneasthecarhasthe \\ tendencytoslidedowntheinclinedplane. \\ Netforcetowardsthecentreis \\ {F}_c=N\sin \theta -\mu N\cos \theta ...1 \\ m\frac{v^2}{R}=N\sin \theta -\mu N\cos \theta \\ Consideringtheequilibriumintheverticaldirection \\ N\cos \theta +\mu N\sin \theta =mg...2 \\ N=\frac{mg}{\cos \theta +\mu \sin \theta } \\ PuttingthevalueofNineqn1 \\ m\frac{v^2}{R}=(\sin \theta -\mu \cos \theta )\times \frac{mg}{\cos \theta +\mu \sin \theta } \\ {v}^2=(\sin \theta -\mu \cos \theta )\times \frac{gR}{\cos \theta +\mu \sin \theta } \\ {v}_{min}=\sqrt{gR\times \frac{(\sin \theta -\mu \cos \theta )}{\cos \theta +\mu \sin \theta }} \\ Regards \end{gathered}$$