# Centripetal force on the surface of earth

Initially, I was looking for how centripetal force is produced on the surface of the rotating earth for a mass kept at any latitude. I went through the following threads -

From there, I understood that the resultant of normal force (N) and gravity (mg) is the required centripetal force. But what is bothering me now is HOW? According to the answers, the normal force is slanted such that it is not exactly opposite to gravity. Thus, they don't cancel out, resulting in a horizontal centripetal force. I'm still confused and have the following questions :

1. Why is normal force slanted in the first place ? (Is it because of the earth's bulge, friction or centrifugal force?)

2. I think there's also a vertically-upwards component of the resultant, why is that? (the resultant of normal force and gravity)  This is what I see, what is the reason behind this?

Source of the image - https://en.wikipedia.org/wiki/Equatorial_bulge

Edit - I've gone through this question Is the normal force equal to weight if we take the rotation of Earth into account? but this doesn't clear my doubt regarding the upward component of the resultant of gravity and the normal force. I posted this question because I wanted some more insight into that poleward force and its relation to the bulge of the earth, which isn't emphasised in that question. kindly reopen my question.

First, some preliminaries. If there is no force on you, you are traveling in a straight line at a constant speed. (Perhaps $$0$$ speed.) If you are traveling in a circle, the total force on you is deflecting you from a straight line into a circle. That total force is centripetal force. The total acceleration is centripetal acceleration.

The total force may be the sum of multiple forces. These forces might be gravity, normal force from the surface of the Earth, friction, and anything else acting on you. So the question is how do these forces arrange themselves so they add up to the total needed to keep you moving in a circle?

Take a look at my answer to Why does a metal ball not trace back its original path if it hits a wall?. It says why the force from a rigid object like the surface of the Earth is divided into two parts. A reaction force perpendicular to the surface is present because the object is rigid. And a friction force along the surface may or may not be present.

Let's start with a simple situation. Suppose the Earth was a perfect sphere and not rotating. Gravity and the reaction force are equal and opposite. You are at rest.

Suppose the Earth was a rotating perfect sphere of ice. At a particular instant, you are on the equator moving at the same speed and direction as the surface. Again the normal force and gravity are perpendicular. The normal force is a little weaker that gravity. The total is just right for you to move in a circle.

If you are not on the equator, gravity and the reaction force don't add up to a force in the right direction to keep you moving in a circle around the axis. If you are moving in a circle, there must be another force. Perhaps you hold on to the surface. Looking at your diagram, what force do you need to make the total add up to the purple arrow?

The reaction force will adjust itself to keep you on the surface. Gravity pulls you onto the surface. The reaction keeps you from going beneath it. Gravity will be a little stronger. You need to pull yourself along the surface in the direction of the blue arrow. If you do, you will stay fixed to one spot on the Earth as it rotates. If you don't you will slide in the direction opposite the blue arrow.

That leaves the question of why the normal reaction force is not aligned with gravity. It has to do with the Earth's bulge.

Earth is not perfectly rigid. The interior is liquid. The ocean is liquid. The crust is solid rock. But over millions of years, rock bends and flows as this example from Eastern Connecticut State University shows. See Structural Geology Photos - Folds

If you have water sitting where you are, it cannot hold on. It will flow toward the equator. Over time, so will rock. It will pile up. Lower latitudes will become fatter. The shape of the surface will change like the diagram shows. When the shape has changed enough that the tilted reaction forces provides needed blue component, there is no further pull toward the equator. • As we know: when the solar system formed all the clumps of matter that would go on to form the planets started out as protoplanetary disks. The proto-Earth contracted from disk towards spherical shape in a process of dissipation of rotational kinetic energy. In the process of contraction gravitational potential energy is released, and converts to rotational kinetic energy. The process of contraction halts at the point where further contraction would require more input of rotational kinetic energy than is released by the contraction.. Jun 10 at 6:53
• I added the comment about contraction from proto-planetary disk because of the following: in any tabletop demonstration or amusement park ride: we see an increase of radial distance as a system is given rotational velocity. Explaining the Earth's equatorial bulge in terms of 'lower latitudes will become fatter' will be the first thing that comes to mind. So there's a bit of a dilemma there. The actual Earth did not get to its current equatorial bulge from a spherical starting point. But then: the OP is already struggling badly, so I can see the need to keep the explanation very accessible. Jun 10 at 7:03
• Thanks for the answer. There are a few things which I'm unclear about. First of all, why "The normal force is a little weaker than gravity." If both gravity and normal are perpendicular to the equator, why would normal be slightly lesser than gravity? Jun 10 at 9:59
• Also, kindly elaborate on this part - "If you are moving in a circle, there must be another force. Perhaps you hold on to the surface." Jun 10 at 10:00
• "Looking at your diagram, what force do you need to make the total add up to the purple arrow?" - I think we would need a force equal in magnitude and opposite in direction to the yellow vector, that way we'll only be left with purple vector i.e, centripetal force as a resultant. Is this right? Jun 10 at 10:17