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Banking of Road: Motion Of Car On Circular Track, Maximum and Minimum Speed Of Safe Turn

While driving a car or bicycle we tend to lower the speed when there is a turn on the road. This is because while taking a turn there is a chance of tires slipping at high speed. So our brain gets the intuition and we automatically apply brakes to reduce the speed. You must have observed that during the rainy season we have to drive slower than usual at turning points. This is because the friction force that helps to make that turn gets reduced. In this article we will study the motion of cars on circular tracks and banking of roads. 

Table of Contents

  • Turning on the horizontal roads
  • Banking of road (without friction)
  • Banking of road (with friction)
  • Practice problems
  • FAQs

Turning on The Horizontal Roads

You must have seen the speed limit written just before taking a turn on these roads. Have you ever wondered how they calculate this speed limit? Let’s analyse the motion of a car taking a turn on a horizontal track. 

Case1: If track is frictionless ()

For a body to undergo circular motion centripetal force is necessary. On a frictionless horizontal track there is no centrifugal force. So we can not take a turn on a horizontal frictionless surface. Still we try to make a turn in that case the car will slip and get out of track. 

Case2: Track is rough ()

In this case friction force will act as centripetal force. The frictional force is static in nature as there is no relative slipping. Let’s calculate maximum speed for the safe turn on horizontal circular rough surface whose coefficient friction is and radius is R. The direction of friction force will be inwards. 


In vertical direction, Normal force is balanced with weight

So,

In horizontal direction,

There the maximum speed for safe turn on horizontal track is .

Banking of Road (without friction)

Banking of roads is the phenomenon in which the surface of a curved road is inclined with the horizontal to provide the necessary centripetal force for taking a safe turn.

Let a car of mass m moves with speed v along a curved road of the radius of curvature R, whose edges are raised by an angle θ with the horizontal. In this case the horizontal component of normal force will act as centripetal force. Look at the diagram given below.


Balancing the forces in vertical direction, …..(1)

In horizontal direction net force towards centre is  

This net force towards centre act as centripetal force which is equal to

Therefore, ………………………..(2)

By taking the ratio of equation (2) and (1), we get

This is the safe speed of the car when the banked road is frictionless. 

Note: 

  • If the speed is more than the safe speed, then the car tends to skid outwards.
  • If the speed is less than the safe speed, then the car tends to skid inwards.

Banking of Road (with friction)

In this scenario the direction of friction will depend on the speed of the car. Because of that we get the range of safe speed. Let a car of mass m move with speed v along a curved road of radius of curvature R with banking angle θ. Since the friction force along the radial direction is a self-adjusting force, it increases or decreases with the speed of the car to avoid slipping. However, this friction has a maximum limit that is known as limiting friction.

Case1: Calculation of Maximum speed limit ()

The velocity of the car, corresponding to the limiting value of friction, is the maximum velocity with which a car can move without slipping. In this case the direction of friction will be downwards along the surface of the banked road because the car has the tendency to slip upward.


Balancing forces in vertical direction,

………(1)

In horizontal direction net force towards centre is

This net force towards centre act as centripetal force which is equal to

Therefore ……(2)

…………………………(3)

By solving these three equations we get,

Case2: Calculation of Minimum speed limit ()

In this case the direction of friction will be upwards along the plane of the banked road as the car has a tendency to slip downward.


Balancing forces in vertical direction,

……………. (1)

In horizontal direction net force towards centre is

This net force towards centre act as centripetal force which is equal to

Therefore ………….(2)

……………………………….(3)

By solving three equations we get,

The range of speed with which a car can move along a circular banked road with friction without slipping is as follows: 

Video Explanation: Banking of roads (time stamp: 1.12.54 to 1.28.36)

Practice Problems

Q 1. What should be the angle of banking of a circular track of radius that is designed for cars with an average speed of ?

Solution: 

.

Q 2. A circular road of radius has the angle of banking equal to . At what speed should a vehicle go on this road so that the friction is not used?

Solution:

Q 3. A circular road of radius has a banking angle of . What will be the maximum safe speed of a car whose mass is , and the coefficient of friction between the tyre and the road is ?

Solution:

Q 4. A circular road of radius has a banking angle of . What will be the minimum safe speed of a car? (coefficient of friction between the tyre and the road is )?

Solution: 

FAQs

Q 1. What is the need of banking of roads?
Solution: Banking of roads is the phenomenon in which the surface of a curved road is inclined with the horizontal to provide the necessary centripetal force for taking a safe turn. The component of normal force helps in turning the car even if the friction becomes zero during rainy seasons. 

Q 2. What is the example of banking of roads?
Solution: Death Well used in the circus is an excellent example of application of banking of roads. 

Q 3. What is the role of friction on the horizontal curved surface?
Solution: Friction provided the necessary centrifugal force to take a safe turn on a curved surface. Without the friction we can take turns on a horizontal curved surface. 

Q 4. What would be the motion of a car if the speed limit exceeds while moving on a curved surface?
Solution: If the speed of the car exceeds its limit then the car will start skidding in an outward direction. 

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