<< Chapter < Page Chapter >> Page >

Skidding down the bank

In this case, the velocity of the vehicle is less than threshold speed " r g tan θ ". Friction acts "up" across the bank. There are three forces acting on the vehicle (i) its weight "mg" (ii) normal force (N) due to the bank surface and (iii) static friction " f s ", acting up the bank. The free body diagram is as shown here.

Banking of road

The speed is less than the threshold speed.

F x N sin θ f S cos θ = m v 2 r

F y N cos θ + f S sin θ = m g

Skidding up the bank

In this case, the velocity of the vehicle is greater than threshold speed. Friction acts "down" across the bank. There are three forces acting on the vehicle (i) its weight "mg" (ii) normal force (N) due to the bank surface and (iii) static friction " f s ", acting down the bank. The free body diagram is as shown here.

Banking of road

The speed is greater than the threshold speed.

F x N sin θ + f S cos θ = m v 2 r

F y N cos θ f S sin θ = m g

Maximum speed along the banked road

In previous section, we discussed various aspects of banking. In this section, we seek to find the maximum speed with which a banked curve can be negotiated. We have seen that banking, while preventing upward skidding, creates situation in which the vehicle can skid downward at lower speed.

The design of bank, therefore, needs to consider both these aspects. Actually, roads are banked with a small angle of inclination only. It is important as greater angle will induce tendency for the vehicle to overturn. For small inclination of the bank, the tendency of the vehicle to slide down is ruled out as friction between tyres and road is usually much greater to prevent downward skidding across the road.

In practice, it is the skidding "up" across the road that is the prime concern as threshold speed limit can be breached easily. The banking supplements the provision of centripetal force, which is otherwise provided by the friction on a flat road. As such, banking can be seen as a mechanism either (i) to increase the threshold speed limit or (ii) as a safety mechanism to cover the risk involved due to any eventuality like flattening of tyres or wet roads etc. In fact, it is the latter concern that prevails.

In the following paragraph, we set out to determine the maximum speed with which a banked road can be negotiated. It is obvious that maximum speed corresponds to limiting friction that acts in the downward direction as shown in the figure.

Banking of roads

The horizontal component of normal force and friction together meet the requirement of centripetal force.

Force analysis in the vertical direction :

N cos θ - μ s N sin θ = m g N ( cos θ - μ s sin θ ) = m g

Force analysis in the horizontal direction :

N sin θ + μ s N cos θ = m v 2 r N ( sin θ + μ s cos θ ) = m v 2 r

Taking ratio of two equations, we have :

g ( sin θ + μ s cos θ ) ( cos θ - μ s sin θ ) = v 2 r v 2 = r g ( sin θ + μ s cos θ ) ( cos θ - μ s sin θ ) v = { r g ( tan θ + μ s ) ( 1 - μ s tan θ ) }

Bending by a cyclist

We have seen that a cyclist bends towards the center in order to move along a circular path. Like in the case of car, he could have depended on the friction between tires and the road. But then he would be limited by the speed. Further, friction may not be sufficient as contact surface is small. We can also see "bending" of cyclist at greater speed as an alternative to banking used for four wheeled vehicles, which can not be bent.

The cyclist increases speed without skidding by leaning towards the center of circular path. The sole objective of bending here is to change the direction and magnitude of normal force such that horizontal component of the normal force provides for the centripetal force, whereas vertical component balances the "cycle and cyclist" body system.

Banking of roads

The horizontal component of normal force meets the requirement of centripetal force.

N cos θ = m g

N sin θ = m v 2 r

Taking ratio,

tan θ = v 2 r g v = ( r g tan θ )

Questions & Answers

A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply

Get the best Physics for k-12 course in your pocket!





Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask