Questions
Draw a diagram showing all components of forces acting on a vehicle moving on a curved banked road.
Draw a neat labelled diagram showing the various forces and their components acting on a vehicle moving along curved banked road
Solution

Motion of a car on a banked road:
For the vehicle to go round the curved track at a reasonable speed without skidding, the greater centripetal force is managed for it by raising the outer edge of the track a little above the inner edge. It is called banking of circular tracks.
Consider a vehicle of weight Mg, moving round a curved path of radius r, with a speed v, on a road banked through angleθ.
The vehicle is under the action of the following forces:

The weight Mg acting vertically downwards

The reaction R of the ground to the vehicle, acting along the normal to the banked road OA in the upward direction
The vertical component R cos θ of the normal reaction R will balance the weight of the vehicle and the horizontal component R sin θ will provide the necessary centripetal force to the vehicle. Thus,
R cosθ = Mg …(i)
R sinθ = `(Mv^2)/r` .......(ii)
On dividing equation (ii) by equation (i), we get
`(R sin theta)/(R cos theta) = (Mv^2//r)/Mg`
`tan theta = v^2/(rg)`
As the vehicle moves along the circular banked road OA, the force of friction between the road and the tyres of the vehicle, F = μR, acts in the direction AO.
The frictional force can be resolved into two components:

μ R sinθ in the downward direction

μ R cosθ in the inward direction
Since there is no motion along the vertical,
R cos θ = Mg + μ R sinθ ……. (iii)
Let vmax be the maximum permissible speed of the vehicle. The centripetal force is now provided by the components R sinθ and μ Mg cosθ, i.e.,
R sin θ + μ R cosθ = `(Mv_max ^2)/r` .....(iv)
From equation(iii),we have
Mg = R cosθ (1−μ tanθ)…(v)
Again from equation (iv), we have
`(Mv_max ^2)/r`
= R cosθ (μ + tanθ) …(vi)
On dividing equation (iv) by (v), we have
`(v_max ^2)/(gr) = (µ + tan theta)/(1  tan theta)`
`=> v_max = [(g r (µ + tan theta)/(1  µ tan theta)]^(1//2)`
Maximum optimum speed depends on:
1) Radius of the curved path,
2) Coefficient of friction
3) angle on inclination
Regards