I have an example in a textbook that goes like this:
A car travels around a bend which has radius 100 m and is banked at an angle of 20° the horizontal. The car is travelling at a speed of 30 ms-. What is the least possible value of the coefficient of friction if the car does not slip up the slope?
Their solution is to resolve vertically and horizontally to find an two expressions for reaction force and the coefficent of friction.
let F = uR where R is the reaction force and u is the coefficient of friction.
Rcos20 - Fsin20 - mg = 0 Rcos20 - uRsin20 = mg R(cos20-usin20) = mg
Rsin20 + Fcos20 = mv^2/r Rsin20 + uRcos20 = m30^2/100 R(sin20 + ucos20) = 9m
Solving dividing these two equations to remove R and m and then solving gives u = 0.416
I would have attempted the problem a different way by resolving parallel and perpendicular to the slope.
R - mgcos20 = 0 => R = mgcos20
F - mgsin20 = 0 => F = mgsin20 => uR = mgsin20
Solving this gives
uR/R = sin20/cos20 => u = tan20 = 0.364
I don't understand where in my method I have gone wrong and any help would be much appreciated.