Question

What is the banking of roads? Obtain an expression for maximum safety speed of a vehicle moving along a curved horizontal road.

Answer 1

Hint: We can use the idea for a circular motion some force to be providing as a centripetal force to act towards the centre.
We can consider all forces acting on the car which can provide sufficient centripetal force.

Complete step by step answer:
We know the banking of roads $$-$$ To avoid risk of skidding the road surface at a bend is tatted inwards, i.e. the outer side of road is raised above its inner ride. This is called ‘Banking of roads’
We can consider taking a left turn along a road or banked at an angle $\theta$ for a designed optimum speed V. Let M be the man of the car. In general, the force is aching on the car one.
(a) Its weight$\overrightarrow{mg}$, acting vertically drawn.
(b) The functional reactor of the road$\overrightarrow{N}$, perpendicular to the road surface.
(c) The frictional force ${\overrightarrow{f}}_1$ along the inclined surface of the road.
Resolve $\overrightarrow{N}$ and $\overrightarrow{f_1}$ into two perpendicular components.
$N\cos \theta$ Vertically up and ${\overrightarrow{f}}_s\sin \theta$ vertically down, $N\sin \theta$ and ${\overrightarrow{f}}_s\cos \theta$ horizontally towards the center of the circular path.
If $V_{max}$ is the maximum safe speed without skidding, then
$$\begin{gathered} {\displaystyle \frac{m{v}_{max}^2}{t}}=N\sin \theta +f_s\cos \theta \\ =N\sin \theta +{\mu }_{s}N\cos \theta \end{gathered}$$
${\displaystyle \frac{m{v}_{max}^2}{t_0}}=N\left(\sin \theta +{\mu }_s\cos \theta \right)...(1)$
And,
$$\begin{gathered} N\cos \theta =mg+f_s\sin \theta \\ =mg+{\mu }_{s}N\sin \theta \\ mg=N\left(\cos \theta -{\mu }_s\sin \theta \right)...(2) \end{gathered}$$
Dividing ($1$) by ($2$)
${\displaystyle \frac{m{v}_{max}^2}{V_{0}mg}}={\displaystyle \frac{N\left(\sin \theta +{\mu }_s\cos \theta \right)}{N\left(\cos \theta -{\mu }_s\sin \theta \right)}}$
$\therefore {\displaystyle \frac{v_{max}^2}{mg}}={\displaystyle \frac{\sin \theta +{\mu }_s\cos \theta }{\cos \theta -{\mu }_s\sin \theta }}={\displaystyle \frac{\tan \theta +{\mu }_s}{1-{\mu }_s\tan \theta }}$
$V_{max}=\sqrt{{\displaystyle \frac{mg\left(\tan \theta +{\mu }_s\right)}{1-{\mu }_s\tan \theta }}}$
This is the expression for the maximum safety speed on a curved horizontal road.

Note: The friction reaction of the road $\overrightarrow{N}$ perpendicular to the road surface.
And the frictional force ${\overrightarrow{d}}_s$ along the inclined surface of the road.
Other than banked roads in formula races we can see they bend the bike at some angle.