21. Meaning of Banking
A banked curve is tilted at an angle θ relative to the horizontal.
The normal force from the road has:
- A vertical component
- A horizontal component
The horizontal component can provide the required centripetal force.
22. Forces on a Frictionless Banked Curve
The normal force N acts perpendicular to the surface.
Vertical Component
There is no vertical acceleration:
N cos θ = mg
Horizontal Component
The horizontal component provides centripetal force:
N sin θ = mv² / r
23. Derivation of Banking Formula
Divide:
N sin θ = mv² / r
by:
N cos θ = mg
Therefore:
N sin θN cos θ = mv²/rmg
Cancelling N and m:
tan θ = v²r g
Thus:
tan θ = v²r g
The ideal speed is:
v = √(r g tan θ)
The required banking angle is independent of vehicle mass.
24. Banking Formula Conditions
The simplified relation:
tan θ = v²r g
assumes:
- Friction is not required.
- The vehicle follows a circular path.
- The speed remains constant.
- The radius is measured horizontally.
- Vertical acceleration is zero.
25. Solved Example: Banked Curve
Question: A frictionless curve has radius 50 m and is designed for a speed of 15 m s⁻¹. Find its banking angle. Take g = 9.8 m s⁻².
Solution
tan θ = (15)² / ((50)(9.8))
tan θ = 225490
tan θ = 0.459
θ = tan⁻¹(0.459)
θ = 24.7°
