Question

A road is banked at an angle of $\theta ={30}^{\circ }$ and provides a coefficient of static friction ${\mu }_s=0.4$. The road gets seriously damaged if it experiences a normal reaction force of more than $10000\text{N}$. What should be the maximum mass of the vehicle allowed on this road?

• A. $666\text{kg}$
• B. $1000\text{kg}$
• C. $5000\text{kg}$
• D. $555\text{kg}$

Answer 1

The correct option is A $666\text{kg}$

From the FBD of vehicle:

For finding maximum mass, we have to take limiting value of friction i.e $f={\mu }_{s}N$

For equilibrium of vehicle in Y direction:
$N\cos \theta -f\sin \theta =mg$
$\Rightarrow N\cos \theta -{\mu }_{s}N\sin \theta =mg$
$\Rightarrow$$N=\frac{mg}{\cos \theta -{\mu }_s\sin \theta }$

Given, maximum tolerable value of normal reaction is
$N_{max}=10000\text{N}$

$\therefore 10000=\frac{m_{max}\times 10}{\cos {30}^{\circ }-(0.4\times \sin {30}^{\circ })}$
$\Rightarrow m_{max}=\frac{10000\times 0.666}{10}=666\text{kg}$