Normal force not perpendicular to the surface In my class about mechanics i had to solve this problem, but it was never really explained. The solution is found beneath in a picture. In the solution they also calculate the angle of the normal force, but isn't the normal force always perpendicular to the contact surface in the contact point, so why isn't the normal force facing outwards and perpendicular to the tangent in that point. I've asked a lot of other students in my class, and none of them seem to understand this.

 A: "Normal force" is simply bad notation. "Contact force" would be better. Usually, we don't distinguish, because the contact force is almost normal to the surface. But in the context of this detailed examination of the rotating Earth, it is confusing not to distinguish!
Later Additions (incorporating comments)
The contact force can be resolved into a component normal to the Earth (modelled as a sphere) and a small tangential (or frictional) component. If this component wasn't present, the body would be slipping round the Earth's surface, towards the equator!
I can't resist remarking that the 'textbook' treatment reproduced in the question is terribly long-winded. The results can be obtained in three or four lines by applying the cosine formula and the sine formula to a simple vector triangle. 
A: I think they just use incorrect terminology. What they call "normal force" is actually a vector sum of the normal force and the force of friction (the body would not rest in an arbitrary point of the surface of rotating Earth without friction). 
