Question

A curve on a highway has a radius of curvature $40\text{m}$. The curved road is banked at an angle of ${45}^{\circ }$ with the horizontal. If the coefficient of static friction that the road provides is ${\mu }_s=0.20$, what is the maximum permissible speed (in $\text{m/s}$) limit for vehicles travelling on the road?
Obtain the answer as nearest integer value
Useful data : $g=10{\text{m/s}}^2$ ,$\sqrt{6}=2.40$

Answer 1

For the case of maximum safe speed on banked rough road i.e $v_{max}$, friction will act DOWN the plane:


Solving equations of dynamics, we get:

$N\sin \theta +f_{max}\cos \theta =\frac{m{v_{max}}^2}{r}.......\left(i\right)$

where $f_{max}={\mu }_{s}N.....\left(ii\right)$

$N\cos \theta -f_{max}\sin \theta =mg.....\left(iii\right)$

Using $\left(i\right),\left(ii\right),\left(iii\right)$

$v_{max}=\sqrt{\left(\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }\right)rg}$
$=\sqrt{\left(\frac{0.20+\tan {45}^{\circ }}{1-0.20\times \tan {45}^{\circ }}\right)40\times 10}$
$\therefore v_{max}=10\sqrt{6}=24\text{m/s}$