Sometimes a cant occurs at a race track or subway line, in order to reduce the g-forces from the side that occur in that bend. Such a cant is, of course, calculated on the basis of the speed of the vehicle. What is this formula? At an infinitely high speed, what is that cant so that no g-forces to the side occur?
Answer
When you take a turn you experience a pseudo force outwards. The magnitude of that pseudoforce is equal to mv2/R, where m is your mass, v is your speed and R is the radius of the turn. Take a look at the attached figure : we decompose the gravitational force (directed vertically downwards) and the centrifugal force (directed horizontally to the left) into a component perpendicular to the orbit, and parallel to the orbit. We want the person to just “feel” a force perpendicular to the track, so that they don’t slide left or right in their seat.
You see the slope of orbit at a certain angle β, which is indicated everywhere in yellow. The gravitational force is the red arrow pointing downwards, the centripetal force is the red arrow pointing to the left. For the components parallel to the slope of the road we find:
gravitation (green arrow pointing to the inside of the bend) : mgsin(β)
average force (blue arrow to the outside of the bend) : mv2/R cos(β)
If a passenger is only allowed to experience a force perpendicular to the slope of the bend, the above two components must cancel each other out:
this eventually gives : g . tan(β) = v2 / R
What if v is infinite? then, in order to maintain the equilibrium, the left-hand side in the above formula must also be infinite, and this is only possible if the angle β is therefore 90°, because the tangent is infinite.
Answered by
prof.dr. Paul Hellings
Department of Mathematics, Fac. IIW, KU Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/
.