A car is negotiating a curved road of radius R. The road is banked at an angle θ. the coefficient of friction between the tyres of the car and the road is μ_s. The maximum safe velocity on this road is:

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AIIMS BSc NURSING 2024 Memory-Based Paper

\(\sqrt{g R^{2} \frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}}\)
\(\sqrt{g R \frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}}\)
\(\sqrt{\frac{g}{R} \frac{\mu_{s}+\tan \theta}{1-\mu_{2} \tan \theta}}\)
\(\sqrt{\frac{g}{R^{2}} \frac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}}\)

Answer 1

Calculation

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From the above diagram, we have

N = mg cosθ + (mv² / R) sinθ (1)

f_max = μN ⇒ f_max = μ_s mg cosθ + (μ_s mv² / R) sinθ

mg sinθ + f_max = (mv² / R) cosθ (2)

Putting the value

mg sinθ + μ_s mg cosθ + (μ_s mv² / R) sinθ = (mv² / R) cosθ

g sinθ + μ_s g cosθ = (v² / R)(cosθ − μ_s sinθ)

gR [ (tanθ + μ_s) / (1 − μ_s tanθ) ] = v²

v = √[ gR (tanθ + μ_s) / (1 − μ_s tanθ) ]