Final answer:
The calculation of the ideal banking angle for a curve involves analyzing the forces and using the principles of circular motion. The formula derived for the ideal banking angle takes into account the vehicle's speed, curve radius, and gravitational acceleration, resulting in the expression θ = tan^(-1)(v^2/(gr)). This angle allows the vehicle to navigate the curve at a specific speed without friction.
Step-by-step explanation:
The derivation of the expression for the banking angle requires an understanding of the forces acting on a vehicle that is traveling through a banked curve without the aid of friction. The goal is to find the angle at which the road should be banked so that the horizontal component of the normal force exerted by the road provides the necessary centripetal force to keep the vehicle moving in a circle at a certain speed. The derivation begins with the understanding that, in the case of an ideally banked curve, the net force causing the centripetal acceleration is the horizontal component of the normal force.
Derivation of the Banking Angle Formula
Let v be the speed of the vehicle, r the radius of the curve, θ the banking angle, g the acceleration due to gravity, and N the normal force exerted by the road. The centripetal force Fc required for circular motion is given by Fc = mv2/r, where m is the mass of the vehicle. The normal force has two components: a vertical component Nv that balances the weight of the vehicle (mg) and a horizontal component Nh that provides the centripetal force needed for the turn.
The horizontal component Nh can be defined as Nh = N × sin(θ), and the vertical component Nv as Nv = N × cos(θ). Since Nv = mg, we can say that N = mg / cos(θ). Substituting into the equation for Nh, we find that Nh = (mg / cos(θ)) × sin(θ). Setting this equal to the centripetal force, we obtain mv2/r = (mg / cos(θ)) × sin(θ).
By simplifying, the mass m cancels out, and we get v2/r = g × tan(θ). Solving for the banking angle θ, we obtain the expression θ = tan−1(v2/(gr)). This is the formula for the ideal banking angle that allows a vehicle to turn at a certain speed without relying on friction.