Question

Define angle of banking. Obtain an expression for angle of banking of a curved road and show that angle of banking is independent of the mass of the vehicle.

Answer 1

Expression for angle of banking:

i. The angle made by the surface of road with horizontal surface of road is called angle of banking. It is given by angle θ.

ii. Consider a vehicle of mass m moving with speed v on a banked road banked at an angle θ as shown in the figure.

iii. Let F be the frictional force between tyres of the vehicle and road surface. The forces acting on the vehicle are

a. Weight mg acting vertically downward.

b. Normal reaction N in upward direction through C.G.

The frictional force between tyres of vehicle and road surface can be resolved into,

Fcosθ - along horizontal direction

Fsinθ - along vertically downward direction.

iv. The normal reaction N can be resolved into two components:

a. Ncosθ along vertical direction

b. Nsinθ along horizontal direction.

v. The component Ncosθ balances the weight mg of vehicle and component Fsinθ of frictional force.

\(\therefore\) Ncosθ = mg + Fsinθ

\(\therefore\) Ncosθ - Fsinθ = mg ….(1)

vi. The horizontal component Nsinθ along with the component Fcosθ of frictional force provides necessary centripetal

force \(\cfrac{m\text v^2}r\).

\(\therefore\) Nsinθ + Fcosθ = \(\cfrac{m\text v^2}r\)....(2)

vii. Dividing equation (2) by (1),

The magnitude of frictional force depends on speed of vehicle for given road surface and tyres of vehicle.

viii. Let v_max be the maximum speed of vehicle, the frictional force produced at this speed is given by,

F_max = μ_sN ….(4)

ix. For a curved horizontal road, θ = 0°, hence equation (6) becomes,

V_max = \(\sqrt{\mu_srg}\)......(7)

x. Comparing equation (6) and (7) it is concluded that maximum safe speed of vehicle on a banked road is greater than that of curved horizontal road/level road.

xi. If μ_s = 0, then equation (7) becomes,

At this speed, the frictional force is not needed to provide necessary centripetal force. There will be a little wear and tear of tyres, if vehicle is driven at this speed on banked road. v_o is called as optimum speed.

xii. From equation (8) we can write,

xiii. Equation (9) represents angle of banking of a banked road. Formula for angle of banking does not involve mass of vehicle m. Thus angle of banking is independent of mass of the vehicle.