Resolve real contact forces to explain road banking and its limits.
Method and worked examples
On a flat curve, weight and normal force balance vertically while static friction supplies the inward force rmv2. Static friction need not equal its maximum value. If the needed force exceeds available grip, the assumed circular motion cannot be maintained. Required force grows with speed squared.
On a friction-free bank, the normal force tilts inward. Resolve Ncosθ= mg vertically and Nsinθ=rmv2 horizontally. Dividing gives tanθ=(v2rg), or v=√rgtanθ. Mass cancels. This is one ideal design speed for a particular radius and bank angle; other speeds generally require friction as well.
Example 1
Find the no-friction bank angle for v=14.0ms−1,r=40.0m and g=9.80ms−2.
tanθ=40.0×9.8014.02=0.500, so θ=26.6∘.
Example 2
A 1000kg car rounds a flat 50.0m curve at 10.0ms−1. Find required friction.
F=rmv2=1000×50.0100=2000N horizontally inward.
Watch out:Banking alone supplies the required resultant only at its ideal design speed; other speeds may require friction.
8. Flat and banked curves | Year 12 Physics | Ascent Arcade