Centripetal force on the surface of earth

Question

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 -

1. Which force provides the centripetal acceleration that makes objects on earth's surface rotate about Earth's axis of rotation?
2. Is the normal force equal to weight if we take the rotation of Earth into account?
3. Question about the Normal Force exerted by Planet Earth in relation to centripetal force
4. If the ground's normal force cancels gravity, how does a person keep rotating with the Earth?

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.

Answer 1

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.

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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.

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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.

Answer 2

EDIT: Added an animation at the end that might be easier to follow.

First let's consider a simpler model of a perfect spherical planet rotating about the vertical axis with North at the top.

Diagram 1:

The only real force present is the force of gravity (a) acting inwards. This force is broken down into two components, (g) and (c). (g) is the centripetal force which is just a component of the gravitational force and there is no balancing force for this component, so the force accelerates the particle inwards and this keeps the particle moving in a circle as the planet rotates. This leaves component force (c) which is dissected into two more components (d) and (e) in the diagram below.

Diagram 2:

(d) is the inward weight force and (f) is a balancing reaction force. (e) is a sideways force and if there is insufficient friction, this unbalanced force tends to accelerate everything on the surface towards the equator. If the Earth was a sphere, this sideways force would eventually deform the Earth and any surface liquid tends to pile up at the equator. (The molten rock below the surface ensures the Earth is not a perfectly rigid object)

Eventually an equilibrium point is reached where the sphere becomes oblate as in the diagram below. The oblateness of the Earth is greatly exaggerated (as is the centripetal/centrifugal force) for clarity.

Diagram 3:

Again, the only real force is the force of gravity (a) acting towards the centre. (c) is the inward component normal to the surface that is felt as weight and (f) is the reaction force to (c). Component force (g) is not balanced and provides the inward centripetal acceleration. Since g is fully accounted for as providing the centripetal force, the only forces left are component (c) and reaction force (f). Since these are parallel and both normal to the surface, there is no longer any sideways force on surface particles. You can view the centripetal force simply as a component of the inward gravitational force. You could interpret the centrifugal force as the resultant of forces (a) and (f), but this is difficult to justify, as force (f) is cancelled by force (c). It also worth noting that the reaction force (f) is not really a 'tilted' normal. It is perpendicular or normal to the surface. It is only tilted in the sense that is not parallel to the inward acting gravity force (a). It is also worth noting that something like the Earth that has had millions of years to find equilibrium should be almost the perfect oblate shape required for equilibrium.

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Food for thought: If you build a tall tower using a plumb bob to ensure it is vertical, will the tower be parallel to (f) or (a)?

P.S. While I narrated this from the point of view of starting with a sphere for simplicity, it should be obvious from the points above that if the the Earth starts as an over oblate smarty shaped planetoid, then the sideways forces will tend to contract the smarty to something closer to spherical at the equilibrium point.

Edit: Decided to add this final free body diagram to illustrate the last point. It represents a zoomed in portion of an over oblate planetoid.

Diagram 4:

(a) is the centre pointing gravitational force. (g) is the component of (a) that provides the centripetal force. That leaves component(c) which is subdivided into two more components, (d) which is the downward normal force, which is balanced by the reaction normal force (f) and sideways force (e) which is not balanced and drives surface material towards the pole until the equilibrium oblate shape is achieved.

Just a doubt. In the first diagram why don't you put a component of the normal force from the ground projected in the direction opposite to the centripetal force? – Mark_Bell

The force opposite to the centripetal force (in the non rotating reference frame) is called the centrifugal force. This is a 'fictitious' force and does not appear in an inertial reference frame. If there was a counter force balancing the centripetal force, there would be no net force to keep the particle moving in a circular path. In the reference frame of a non inertial observer rotating with the Earth, the centrifugal force is real.

In the above animation, the initial primary forces are the gravitational force (a) and the centrifugal force (b) as seen by an observer on the surface rotating with the planet. Force a is resolved to component (d) that opposes the centrifugal force and component (c). The centripetal force (d) and the centrifugal force (b) cancel each other and there is no motion parallel to the equatorial plane. This leaves component (c) that is resolved to components (f) normal to the surface and (e) tangential to the surface. There is a reaction force (g) also normal to the surface and this cancels the force (f). The only force left is component (e) tangential to the surface that acts towards the equator. This could be opposed by sufficient friction force, but in the absence of friction (or very little friction such as surface liquid) the representative particle is propelled towards the equator. In the inertial reference frame, the fictitious centrifugal force vanishes and there is no force to oppose the centrifugal force and the particle follows a circular path as seen from the North pole.

The interactive animation is here. The oblateness of the spheroid can be altered with the R slider to find the equilibrium point (around R=5) and the over oblate case can be investigated, where the residual force is towards the pole. The animation is controlled by the video play button at the bottom left or alternatively gone through step by step or in slow motion using the T slider.