Motion of Vehicle on a Plane and a Banked Circular Path Physics Notes

Motion of Vehicle on a Plane and a Banked Circular Path Physics Notes

Motion of Vehicle on a Plane and a Banked Circular Path:
1. A circular turn on a level road: Consider a car of weight ‘mg’ going around on a circular turn of radius Y with velocity v on a level road as shown in the figure. While rounding the curve, the wheels of the vehicle have a tendency to leave the curved path and regain the straight-line path.

(a)

The Force of friction between the wheel and the road opposes this tendency of the wheel. Therefore, this frictional force acts towards the centre of the circular path and provides the necessary centripetal force.

Three forces are acting on the car Figure.

1. The weight of the car, mg, acting vertically downwards,
2. Normal reaction N of the road on the car, acting vertically upwards,
3. Frictional force F, along the surface of the road, towards the centre of the turn,

As there is no acceleration in the vertical direction,
N – mg = 0
or N = mg …(i)
Since, for safe driving of a car, on the circular path, the centripetal force must be equal to or less than friction force.

Here, μ is the coefficient of static friction between the tyres and the road.
Hence, the maximum velocity with which a vehicle can go round a level curve; without slidding is
v = \(\sqrt{μ r g}\)

The value of v depends upon:

• Radius r of the curve.
• Coefficient of friction (μ) between the tyre and the road.

Clearly, v is independent of the mass of the car

2. Banking of roads: The value of maximum velocity for a vehicle to take a circular turn (without skidding) on a level road is quite low. This limiting value of the velocity decreases further due to a decrease in the value of the coefficient of friction p on a slippery road and for a vehicle, whose tyres have worn out. Therefore especially in hilly areas, where the vehicle has to move constantly along the curved tracks, the maximum speed at which it can be run, will be very low.

If any attempt is made to run it at greater speed, the vehicle is likely to skid and go off the track. In order that the vehicle can go round the curved tracks at a reasonable speed without skidding, sufficient centripetal force is managed for it by raising the outer edge of the track a little above the inner one. It is called the banking of the circular tracks.

Consider a vehicle of weight ‘mg’ moving around a curved path of radius r with speed v on a road banked through angle θ. If OA is a banked road and OX is a horizontal line, then ∠ AOX = θ is called the angle of banking, (figure).

(a)

(b)

(c)
The vehicle is under the action of the following forces:

1. Weight ‘mg’ of the vehicle acting vertically downwards.
2. Normal reaction N of the ground to the vehicle acting along normal to the banked road OA in the upward direction.
3. Force of friction ‘F’ between the banked road and the tyres, acting along with OA

‘N’ can be resolved into two rectangular components:
(a) N cosθ, along the vertically upward direction.
(b) N sinθ, along the horizontal, towards the centra of the curved round.

‘F’ can also be resolved into two rectangular components:
(a) Fsinθ, along the vertically downward direction
(b) Fcosθ, along the horizontal, towards the centre of the curved road.

As there is no acceleration along the vertical direction, the net force along this direction must be zero.
Therefore, N cos θ = mg + Fsinθ …(1)
The horizontal component N sin0 and Fcos0 will provide the necessary centripetal force to the vehicle,
Thus N sin θ + F cos θ = \(\frac{m v^{2}}{r}\) ……….(2)

But F ≤ μ_s N, where μ_s is the coefficient of static friction between the banked road and tyres. To obtain v_max, we put F = μ_s N in equations (1) and (2)


Equation (6) represents the maximum velocity of vehicle on a banked road.

Discussion:
1. If μ_s = 0 i.e., if the banked road is perfectly smooth, then from equation (6)
v_max = (rgtanθ)^1/2
This is the speed at which a banked road can be rounded even when there is no friction. Driving at this speed on the banked road will cause no wear and tear of the tyres.
From equation (7) v² max = rgtanθ
or tanθ = \(\frac{v_{\max }^{2}}{r g}\) …(8)

2. If h is the height ABof outer edge of the road above the inner edge and l = OA is breadth of the road then from the figure.


Usually h <<< l, therefore h² is negligibly small compared to l².
then, tanθ = \(\frac{h}{l}\) …….(9)
From equation (7) and (9)
tanθ= \(\frac{v_{\max }^{2}}{r g}=\frac{h}{l}\) …(10)

Roads are usually banked for the average speed of vehicles passing over them. However, if the speed of a vehicle is somewhat less or more than this, the self-adjusting static friction will operate between the tyres and the road, and the vehicle will not skid.

On the same basis, curved railway tracks are also banked. The level of the outer rail is raised a little above the level of the inner rail while lying on a curved railway track.

Physics Notes