# What forces are involved with Earth's rotational bulge?

• AntiElephant
In summary: This difference in velocity creates a difference in centrifugal force.In summary, the Earth bulges at the equator due to its rotation and the centrifugal force that is created as a result. This force partially offsets the gravitational force, leading to a bulging effect. This can be explained by considering a point on the equator and one on the poles, and drawing a free body diagram for each. The difference in velocity between the equator and poles creates a difference in centrifugal force, which leads to the bulging effect. This is a complex problem that requires consideration of the direction of force vectors and the gravitational potential, especially when taking into account variations in density with depth.
AntiElephant
I'm trying to understand mathematically, if possible, why it is that the Earth bulges at the equator as a result of its rotation and how exactly gravity manages to keep it all together. Would the better approach be to keep myself in a rotating frame of reference? I lack some knowledge of Netwon's Laws in non-inertial frames of reference but maybe just enough to understand what is going on here.

I want to focus on a point (a "piece" of Earth's matter) on the Earth's surface, along the equator, to understand why it bulges outwards. If the Earth was initially stationary and spherical, then the only force acting on this piece would be gravity $F_{grav}$. As the Earth gradually begins to rotate a centrifugal force $F_{centrif}$ appears pointing in a direction outwards, opposite to the axis of rotation, a fictitious force as a result of being in a non-inertial frame of reference.

The total force on this piece would be $F_{grav} - F_{centrif}$. If $F_{centrif} <= F_{grav}$ then surely the piece would still have a resulting force pointing towards the centre of the Earth and the Earth would remain spherical? If/once $F_{centrif} > F_{grav}$ then the piece would "fly" off from the Earth (in a stationary frame, this would be a result of inertia). Have I looked at this too simplistically? How, then is it that the Earth bulges and instead isn't at either of the extremes - either spherical or stuff "flying" off as a result of inertia?

You can't do this without considering directions of force vectors.
Draw a free body diagram for a point on the equator, another on one of the poles, and one in between.
The key is that the centrifugal and gravitational forces each point somewhere else.

I think the key that is being missed here is this:
AntiElephant said:
If the Earth was initially stationary and spherical, then the only force acting on this piece would be gravity $F_{grav}$.
If a mass has only one force acting on it, it accelerates. So if the Earth is stable/stationary and gravity is pulling a mass down, what is pushing it up to keep if from falling to the center of the earth? What's the other force you are missing?

A.T.
You know, this is hardly a trivial problem, and especially so, if we did not initially know that the gravitating body was an oblate spheroid. Deriving the gravitational potential requires a fairly involved integral. The direction of the gravitational acceleration is not directly away from the center of mass.

Last, if we want the solution to include a variation in density with depth, we need to establish equipotential surfaces in the interior of the spheroid.

Last edited:
Simplest explanation:
The poles of a spinning sphere are stationary except for rotation. The equator is the most rapidly rotating. The rapid rotation partially offsets gravity via centrifugal force. This tapers from equator to pole.

tfr000 said:
The poles of a spinning sphere are stationary except for rotation. The equator is the most rapidly rotating.
The rotation (angular velocity) is the same for both. The radius and linear velocity are different.

## 1. What is the cause of Earth's rotational bulge?

The Earth's rotational bulge is caused by the centrifugal force of the Earth's rotation. As the Earth rotates, the centrifugal force pushes outward, causing the equator to bulge and the poles to flatten.

## 2. How does the Earth's rotational bulge affect its shape?

The Earth's rotational bulge causes it to have an oblate spheroid shape, meaning that it is slightly flattened at the poles and bulging at the equator.

## 3. Does the Earth's rotational bulge impact its gravitational pull?

Yes, the Earth's rotational bulge does have an impact on its gravitational pull. The bulge at the equator causes the planet to have a slightly weaker gravitational pull at the equator compared to the poles.

## 4. How does the Earth's rotational bulge affect the length of a day?

The Earth's rotational bulge has a small impact on the length of a day. As the bulge shifts, it causes a slight change in the Earth's rotation, resulting in a difference of a few milliseconds in the length of a day.

## 5. Are there any other forces involved with the Earth's rotational bulge?

Aside from the centrifugal force, there are also other forces involved with the Earth's rotational bulge. The gravitational pull of the moon and sun also play a role in the bulging of the Earth's equator. Additionally, the Earth's internal structure and composition can also impact its shape and rotational bulge.

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