What is banking of road? Angle of banking and expression.

What is banking of road?

Banking of road : For safety of a vehicle moving along a curved road at high speed, the road surface is kept inclined to horizontal so that outer edge of road is at higher level than inner edge. This construction or Carrangement of road is called as banking of road.


Angle of bankingθ ): The angle made by a banked road surface with horizontal is called as the angle of banking.


Expression for banking of road ( θ ) : 

1) Consider a vehicle of mass 'm moving with speed v, along a banked road. Let 'r' be the radius of curvature and θ be the angle of banking.

2) The forces acting on the vehicle are, when vehicle is moving a banked road are__

i) The weight mg acting in a vertically downward direction.

ii) The normal reaction N perpendicular to the banked road.

3) The normal reaction N can be resolved into two components,

i) N cosθ in vetically upward direction.

ii) N sinθ along horizontal direction.



Where
C.G. = Centre of gravityof the vehicle 
 θ = angle of banking 
m = mass of vehicle 
mg = weight of the vehicle 
N = normal reaction of road surface 
N cosθ = vertical component of the normal reaction.
N sinθ = horizontal component of the normal reaction.

4) Now, N cosθ balances the weight (mg) of vehicle,

N cosθ = mg 

5) N sinθ provides the necessary centripetal force for circular motion.

$\sin \theta=\frac{m v^{2}}{r}$

6) By dividing equation (2) by (1), we get

$\frac{N \sin \theta}{N \cos \theta}=\frac{m v^{2} / r}{m g}$

$\therefore$
$\tan \theta=\frac{\operatorname{mv}^{2}}{r \mg}$
$\therefore \quad \tan \theta=\frac{\mathrm{v}^{2}}{\mathrm{rg}}$
$\theta=\tan ^{-1}\left(\frac{\mathrm{v}^{2}}{\mathrm{rg}}\right)$


This is the expression for angle of banking The angle of banking is independent on mass of vehicle as the term 'm' is absent. 

Equation (3) can be written as,

$\begin{aligned} & v^{2}=\operatorname{rg} \tan \theta \\ \therefore \quad V &=\sqrt{\operatorname{rg} \tan \theta} \end{aligned}$

This is the expression for maximum speed with which a vehicle can be safely, driven along the banked road. It is also clear that the maximum velocity does not depend upon mass of a vehicle (m). 

Maximum velocity depends upon -
i) angle of banking (θ)
ii) radius of curved road (r)
iii) acceleration due to gravity at that place (g).

8) From equation (3), the factors which decide the value of angle of banking are,
i) velocity of vehicle.(v)
ii) radius of curved road (r)
iii) acceleration due to gravity at that place (g)


Necessary of banking of road

(Need for banking of road)

1) When a vehicle moves along horizontal curved road, necessary centnpetal force is supplied by the force of friction between the wheels of vehicle and surface of road.

2 Frictional force is not enough and reliable every time as it changes when road becomes Oily or wet in rainy season.

3) To increase the C.PE. the road should be made rough. But it will cause wear and tear of the tyres of the wheel.

4) Thus, due to lack of centripetal force vehicle tends to skid.

5) When the road is banked, the horizontal component of the normal reaction provides the Necessary centripetal force required for circular motion off vehicle. 

6) To Provide the necessary centripetal force at the curved road banking of road is necessary.



MCQs on Banking of road 

1. While taking turn on a curved road, a cyclist has to bend through a certain angle. This is done______
(a) to reduce his speed
(b) to decrease the friction between the tyres and the road
(c)To get the necessary centripetal force
(d) to reduce his weight

Answer : C


2. An aeroplane is taking a turn in a horizontal plane. While
taking the turn_____

(a) it remains horizontal
(b) it inclines outwards
(c) it inclines inward
(d) it makes its wings vertical

Answer : C


3. When a car takes a turn on a horizontal road, the centripetal force is provided by the______
(a) weight of the car
(b) normal reaction of the road
(c) frictional force between the surface of the road and the tyres of the car
(d) centrifugal force

Answer: C


4. While taking a sharp turn, a car moving on a horizontal road, may be thrown out of the road. This happens_______
(a) due to frictional force between the tyres and the road
(b) due to gravitational force
(c) due to lack of sufficient centripetal force
(d) due to the reaction of the ground.

Answer: C


5.When a car takes a circular turn on a banked road, the Centripetal force is provided by______
(a) gravitational force
(b) frictional force
(c) If horizontal component of normal reaction
(d) vertical component of normal reaction.

Answer: C


6. A cyclist moves in a circular track of radius 100m. If the coefficient of friction is 0.2, then the maximum velocity with which the cyclist can take the turn with leaning inward Is_______

(a) 140 m/s

(b) 14 m/s

(c) 1.4 m/s

(d) 4.9 m/s

Answer:  B


7. A curved road having a radius of curvature of 30m is banked at the correct angle. If the speed of the car is to be doubled, then the radius of curvature of the road should be__________

(a) 62 m

(b) 120 m

(c) 90 m

(d) 15 mn

Answer: B


8. A vehicle sometimes overturns while taking a turn. When it overturns______

(a) The outer wheels leave the ground first

(b) The inner wheels leave the ground first

(c) All wheels leave the ground simultaneously

(d) either the inner wheels or the outer wheels leave the ground first. It depends upon the total weight of the vehicle

Answer: B


9. A car is moving in a circular horizontal track of radius 10m with a constant speed of 10 m/s. A plumb line is suspended from the roof of the car by a light rod of length 1m. What is the angle made by the rod with the track? (g = 10 m/s)

(a) 30°

(b) 45°

(c) 60°

(d) 0°

Answer: B


10. On a banked road, the centripetal force is provided by________

(a) the frictional force

(b) weight of the car.

(c) vertical component of the (resultant) normal reaction

Answer: B