Question

If circular race track is banked at angle $\theta$ ($\theta >$ angle of repose),radius of track is $r$ and $\mu$ is coefficient of friction between car and road. Calculate minimum speed on car on inclined plane to avoid slipping?

• A. $v=\left[rg\left(\frac{\tan \theta -\mu }{1+\mu \tan \theta }\right)\right]$
• B. $v={\left[rg\left(\frac{\tan \theta +\mu }{1-\mu \tan \theta }\right)\right]}^{1/2}$
• C. $v={\left[rg\left(\frac{\tan \theta -\mu }{1+\mu \tan \theta }\right)\right]}^{1/2}$
• D. $v={\left[rg\left(\frac{\tan \theta +\mu }{1+\mu \tan \theta }\right)\right]}^{1/2}$

Answer 1

The correct option is B $v={\left[rg\left(\frac{\tan \theta +\mu }{1-\mu \tan \theta }\right)\right]}^{1/2}$

$Ncos\theta =mg+fsin\theta \to 1\frac{m{v}^2}{r}=Nsin\theta +fcos\theta \to 22÷1\frac{\frac{m{v}^2}{r}}{mg}=\frac{Nsin\theta +fcos\theta }{Ncos\theta -fsin\theta }f=\mu N\frac{v^2}{rg}=\frac{Nsin\theta +\mu Ncos\theta }{Ncos\theta -\mu Nsin\theta }v=\sqrt{rg\left(\frac{tan\theta +\mu }{1-\mu tan\theta }\right)}$