# Reason for two tides on Earth

## Main Question or Discussion Point

There was a thing that kept me wondering for several years, which no science teacher managed to explain. Why does gravitational pull of the Moon causes two tides on Earth at the same time? Would'nt it be more logical for the tide to only occur directly under the moon, nowhere else, because gravity of the Moon is the strongest at that point? Why does the tide also occur where it's the furthest on Earth from the Moon?

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A.T.
It is the non uniform gravity. The Moon's gravity pulls the Earth's center stronger than the Earth's opposite side. So the Earth's center is pulled away from the Earth's opposite side. The same happens with the Moon, due to Earth's gravity. Here explained with vectors:

Some people invoke the inertial centrifugal force, due to orbiting around a point that is not the center of the Earth. But that is a misleading way to explain it. The tidal force would be the same in linear free fall at the same distance:

http://www.vialattea.net/maree/eng/index.htm

Also note the ocean tides on the Earth are very different than the idealized tidal effect. Mostly due to land masses:

http://www.uwgb.edu/dutchs/EarthSC202Notes/TIDES.HTM (Ignore Fig.1 It is the wrong explanation mentioned above.)

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Simon Bridge
Homework Helper
That's neat - I had heard a short qualitative description that says there are two bulges because, though the water to one side, it also pulls the earth. But the above is better.

Not saying this is the correct reason, but an interesting take on it.....

The moon and the earth can be seen as orbiting around each other, and will therefore have an orbital period which is defined by the distance of their centres of gravity. Now it will only be those points which are actually orbiting at the correct distance/velocity/period, all other points will be orbiting at an incorrect velocity for their distance. For solids this does not matter too much as molecular forces take the strain.

However for liquids this is not the case, therefore they drift towards a more acceptable orbit for their velocity (either trying for a longer or shorter orbit) and this is what caused the tides.

(this did not turn into a very good description).

Well, note that gravity force goes down with radius. Closer you are the stronger the force, farther you are the lesser the forces.

So at any moment in time, moon's gravity on earth pulls strongest on the oceans of the earth that are facing it. This causes tides on the closest side to the moon. Next, the earth is also pulled to the moon but less since it is farther away. And finally, as the earth is also pulled away, the water on the other side of the earth is left back. And so the 2nd tide occurs.

Watch Richard Feynman explain it better than me in a short clip:

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D H
Staff Emeritus
Not saying this is the correct reason, but an interesting take on it.....

The moon and the earth can be seen as orbiting around each other, and will therefore have an orbital period which is defined by the distance of their centres of gravity. Now it will only be those points which are actually orbiting at the correct distance/velocity/period, all other points will be orbiting at an incorrect velocity for their distance. For solids this does not matter too much as molecular forces take the strain.

However for liquids this is not the case, therefore they drift towards a more acceptable orbit for their velocity (either trying for a longer or shorter orbit) and this is what caused the tides.

(this did not turn into a very good description).
No. That is essentially the wrong explanation that A.T. argued against in post #2.

Also note that the Moon and Sun induce tides in the Earth itself. These Earth tides are about 1/3 as large as the ocean tides.

So the Earth's center is pulled away from the Earth's opposite side.
But how does Earth's centre being pulled stronger than the furthest of the Earth from the Moon affects the Earth itself, to have the tide to occur on the furthest side? The only reasonable explanation of such forces cancelling out I can think of is Earth falling towards the Moon.

D H
Staff Emeritus
The only reasonable explanation of such forces cancelling out I can think of is Earth falling towards the Moon.
That's exactly right. The Earth is falling towards the Moon. The Earth exerts a gravitational force on the Moon, so by Newton's third law, the Moon exerts an equal but opposite gravitational force on the Earth. That gravitation obeys Newton's third law is built in to Newton's law of gravitation.

Imagine two drops of water, one on the point on the surface of the Earth that is closest to the Moon, the other on the point on the surface of the Earth that is furthest from the Moon. That first drop is slightly closer to the Moon than is the center of the Earth, so the gravitational acceleration at this drop's location toward the Moon is slightly greater than the acceleration of the Earth as a whole toward the Moon. The other drop is slightly further from the Moon than is the center of the Earth, so the gravitational acceleration at this other drop's location toward the Moon is slightly less than that of the Earth as a whole.

Another way to look at it: The Moon pulls the tidal bulge that is closer to the Moon away from the Earth as a whole, and the Moon pulls the Earth as a whole away from the tidal bulge that is opposite the Moon.

university of chicago graduate problems in physics book illustrate the shape of tides induced and it is shown there that it occurs twice by considering the centre of mass of the two as reference.

A.T.
Watch Richard Feynman explain it better than me in a short clip:
He explains it very well via the gravity gradient, but at the and he also mentions the "centrifugal force", as an alternative explanation. Today we understand the "inertial centrifugal force" as a force acting radially in reference frames which axes are rotating. If you just attach the reference frame to the Earths center, but keep the axes orientation constant, you don't have a "inertial centrifugal force" in that frame, just a "inertial uniform force", parallel to the Moon-Earth axis. This the same "inertial uniform force" that is added to the gravity vectors between these two pictures:

Inertial frame, where only gravity exists: Linearly accelerated restframe of the center, where a uniform inertial force is added to gravity: Personally I don't think that invoking that inertial force is a good explanation for the deformation: It is a uniform force so it cannot defom anything.

Of course you can use frames that are actually rotating and have an "inertial centrifugal force". But even in those frames that centrifugal force cannot be blamed for the two bulges. It can partially explain the equatorial bulge in those frames, if the Earth is spinning in the inertial frame. Otherwise it is canceled by the inertial Coriollis force. See:

http://www.vialattea.net/maree/eng/index.htm

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D H
Staff Emeritus
Personally I don't think that invoking that inertial force is a good explanation for the deformation: It is a uniform force so it cannot defom anything.
Why not? You used that inertial force yourself in those images, and in your words!

That inertial force is, to me, the easiest way to explain the tides. One way to explain the tides is to express the tidal forces in an Earth-centered frame of reference. That's an accelerating reference frame. With this explanation, the tides result from the Moon's non-uniform gravitational field (gravity gradient) and from the acceleration of the Earth as a whole toward the Moon (inertial force). That is exactly what your two images depict.

You can explain the tidal bulge from the perspective of any frame of reference. All frames of reference are equally valid after all. However, it is not easy to do this correctly from the perspective of an Earth-Moon barycenter frame, rotating or non-rotating. The explanations that rely on centrifugal force are the canonical example of how to do it incorrectly. That these explanations get the right answer is due to a combination of errors that cancel one another out. Just because two wrongs get the right answer does not mean that it is right.

A.T.
Why not?
To clarify what I meant: "invoking that inertial force as a reason for the deformation is not good". Because as I said: A uniform force field cannot deform anything.

You used that inertial force yourself in those images, and in your words!
My explanation in words actually uses no inertial forces, just the differential gravity: The Moon's gravity pulls the Earth's center stronger than the Earth's opposite side. So the Earth's center is pulled away from the Earth's opposite side. To me the first picture already shows that the center and the opposite side are being pulled apart.

Going into the center's frame (second picture) maybe makes it more obvious visually, but it is still solely the gravity gradient that "causes" deformation, not the uniform inertial force field. Using the inertial force in an explanation is okay, but not as a cause of deformation.

You can explain the tidal bulge from the perspective of any frame of reference. All frames of reference are equally valid after all.
Yes, but since the gravity gradient explains the tidal deformation completely in the inertial frame, it also explains it completely in all other frames (the gravity gradient and the tidal deformation are the same in all frames). So whatever inertial forces you introduce through the choice of the frame, their net effect on the tidal deformation is zero.

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