Question:

A circular racetrack of radius 300 m is banked at an angle of 15^o. If the coefficient of friction between the wheels of a race-car and the road is 0.2 .What is the

• (a) Otimum speed of the race-car to avoid wear and tear on its tyres and
• (b) Maximum possible speed to avoid slipping?

Solution: Optimum and maximum speed of race car

Banked at an angle 0f 15^o

Optimum speed of the race-car:

The horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping.At the optimum speed, the normal reaction’s component is enough to provide the necessary centripetal force and the frictional force is nor needed

$${v=\sqrt{rg\tan\theta}}$$

$${v=\sqrt{(300)(9.8)\tan 15^\circ}}$$

$${v=\sqrt{(2940)(0.2679)}}$$

$${v=\sqrt{787.63}}$$

$${v=28.1 \ \frac{m}{s}}$$

The optimum speed of the race-car to avoid wear and tear is 28.1 ms^-1 .

Maximum possible speed to avoid slipping:

$${v_m=\sqrt\frac{rg(\mu+\tan\theta)}{1-\mu\tan\theta}}$$

$${v_m =\sqrt\frac{(300)(9.8)(\tan15^\circ+0.2)}{1-(0.2)\tan15^\circ}}$$

$${v_m =\sqrt\frac{2940(0.2679+0.2}{1-0.2(0.2679)}}$$

$${v_m=\sqrt\frac{(2940)(0.4679)}{1-0.0536}}$$

$${v_m=\sqrt\frac{1375.63}{0.9464}}$$

$${v_m=\sqrt{1453}}$$

$${v_m =38.1 \ \frac {m}{s}}$$

The maximum possible speed is given by 38.1 ms^-1

EXAMPLES FROM REAL LIFE TO ILLUSTRATE ROAD BANKING

Mountain roads: Mountain roads include strong curves that are banked to minimize accidents, especially in locations with steep slopes.

Highway interchanges: Curved ramps and turns are common features of highway interchanges. These bends are banked to allow cars to pass through them at greater speeds without sliding or overturning.

Banked bends are found on racetracks: The banking helps cars to accelerate through the bends, lowering the chance of an accident and improving performance.

Bridges and elevated roads: Curved parts of elevated highways and bridges are frequently banked to guarantee safe vehicle navigation.

Conclusion:

Banking of road is needed at the turning of a road so that to choose the greatest possible speed of the vehicle, in order to stay away from skidding.and additionally the security of the vehicle moving. Did you at any point really take a look at the most extreme conceivable speed of your vehicle on a banked road?