A road with radius of curvature is 12m is to be banked so that any vehicle can take a bend at 30m/s without ha

A road with radius of curvature is 12m is to be banked so that any vehicle can take a bend at 30m/s without having to rely on sideways friction . Find the angle of banking

======================================…

tan θ = v² /(Rg) = 30²/(12*9.8)
θ = 82 degrees
------------------------------
Road is banked only for a velocity of 30m/s, for vehicles having less or more speed than this cannot depend on the banking alone and has to rely upon fiction. Hence banking alone is not the solution.

draw a diagram of the banked road, with the normal perpendicular (of course) to the road

the vertical component of the normal force must equal the weight of the car, so if we define theta as the angle between the normal and the vertical, we have

N cos (theta) = m g

the horizontal component must equal the centripetal force, so we have

N sin(theta) = m v^2/r

divide the second equation by the first:

sin(theta)/cos(theta) = v^2/rg

since sin/cos = tan, we have

tan(theta) = v^2/r g

therefore, we have:

tan(theta) = (30m/s)^2 /(12m x 9.8m/s/s) = 7.65

theta = 82.6 deg

that seems a little steep, but then a curve of 12m radius is pretty tight to be going 30m/s...