PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
A highway exit ramp is constructed as a smooth, frictionless spiral that maintains a constant banking angle of 8.0∘ while the radius of curvature gradually increases from 60m at the beginning of the ramp to 120m farther along the path. Drivers are expected to reduce speed as they proceed through the ramp.
Assume g=9.8m⋅s−2.
(a) Derive an expression that relates the ideal speed v to the instantaneous radius of curvature r for a vehicle on this ramp, given the fixed banking angle.
(b) Calculate the speed that matches the banking angle at the point where the radius is 60m.
(c) Calculate the corresponding ideal speed after the radius has increased to 120m. Comment on how the change in radius affects the required speed.
(d) If a driver travels more slowly than the ideal speed at the 120m location, state the direction (up or down the slope) in which the static friction force must act on the tires, and explain your reasoning qualitatively.