Understanding the Moon's Role in Tides

[chmate]

Thank you all.

#

The correct explanation was given by Newton in 1687. The Moon's gravity pulls on the Earth and the water on it, but the force of the Moon's gravity varies across of the Earth. The pull is greater on the side facing the Moon, pulling the water there closer to the Moon, while the pull is weaker on the side away from the Moon, making the water there lag behind. This stretches out the Earth and the water on it, creating two bulges. Remember that both the Earth and the Moon are falling towards each other. The reason why they don't collide, is that they already have a motion perpendicular to the direction in which they are falling, so the falling only results in a change in that direction.

http://www.math.nus.edu.sg/aslaksen/teaching/tides.html

Is this explanation correct? My book uses centrifugal force to explain the tides.

Thank you for your contribution.

 

[granpa]

I appear to have made a very basic mistake. the net centrifugal force (for a set of points along the equator due to the Earth's spin and orbital motion around the earth-moon center of gravity) is not the sum of a set of vectors pointing toward the center of the Earth and a set of vectors pointing toward the earth-moon center of mass but appears to be instead the sum of a set of vectors pointing toward the center of the Earth and a set of vectors pointing toward a point at infinity.

the Earth's rotation would then be correctly entirely ignored. all points on the Earth would indeed describe circles with the same radius, but different centers, in the course of the month. the centrifugal force (due to the Earth orbiting the earth-moon center of mass) would be reduced to a constant. and the tides would be due entirely to the difference in the strength of gravity from one side of the Earth to the other.

I stand corrected. my apologies.

DH:your confusing and argumentative comments have not been helpful at all.

and I am very well aware that the surface of the Earth is an equipotential. I am also very well aware that there is an anti-bulge between the moon side bulge and the bulge on the opposite side of the earth. but it was irrelevant to my point.

but its all moot now since my arguments were based on a misunderstanding.

 

[D H]

Thank you all.

http://www.math.nus.edu.sg/aslaksen/teaching/tides.html

Is this explanation correct? My book uses centrifugal force to explain the tides.

Thank you for your contribution.

Yes, although that site is lacking in that it doesn't do the math. There is no need to invoke centrifugal force for the simple reason that centrifugal force does not explain what is going on. Using centrifugal force to explain tides is (pick one or more) the wrong explanation, the wrong math, or the wrong use of the term centrifugal force.

To arrive at the proper explanation all you need to do is compute the apparent gravitational acceleration at the point of interest from the perspective of an Earth-centered reference frame. This is a simple subtraction of the gravitational acceleration of the Earth as a whole toward the Moon from the gravitational acceleration the point of interest (usually a point on the surface of the Earth).

Since the distance between the Earth and Moon is significantly greater than the Earth's radius, first order approximation will yield a very good estimate of this differential gravitational force. Assuming a spherical Earth, the tidal acceleration at some point on the surface is

a_tidal ≅ GM_mr_e/r_m³ (2cosθ xhat - sinθ yhat)

where

• r_m is the distance from the center of the Earth to the center of the Moon
• r_e is the radius of the Earth
• θ is the angle between the vector from the center of the Earth to the center of Moon and the vector from the center of the Earth to the point in question
• xhat is the unit vector along the Earth-to-Moon vector
• yhat is normal to xhat such that the vector from the center of the Earth to the point in question is r_e(cosθ xhat + sinθ yhat).

There is not only a bulge at the sub-Moon point and its antipode, there is also a "squeeze" along the great circle defined by those two points.

 

[DaveC426913]

There is not only a bulge at the sub-Moon point and its antipode, there is also a "squeeze" along the great circle defined by those two points.

There are an infinite number of great circles defined by those two points (the meridia).

I assume you're referring to a great circle that is farthest from both? I'm not sure that two points can define such a locus on anything other than a mathematically perfect sphere. But I am not an expert in this field.

 

[D H]

The simplifying assumption is that the Earth is a perfect sphere, and (oops) the great circle in question is the meridia.