Newton’s Law of Gravitation has made it possible to analyse the tide- producing forces

Undoubtedly Newton’s Law of Gravitation has made it possible to analyse the tide- producing forces. The Law of Gravitation states that every particle of mass in the universe attracts every other particle of mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between the masses.

The greater the mass of the objects and closer they are together, the greater will be the gravitational attraction.

Mathematically this may be formulated as:

Gravitation Force = m₁ m₂ / r2

Where G is the universal gravitational constant, mi and m₂ the masses, and r the distance between the masses. It may be stated that for spherical bodies, all of the masses can be considered to exist at the centre of the sphere. Thus, r will always be the distance between the centers of the bodies to be considered.

Here the tide-generating force must be distinguished from the gravitational attraction. As a matter of fact, the tide-generating force varies inversely as the cube of the distance from the centre of the earth to the centre of the tide-generating object, instead of varying inversely to the square of the distance as in the case of gravitational attraction.

Thus, it is clear that the tide-generating force, even though it is derived from the force of gravitational attraction, is not proportional to it (the gravitational attraction). So for as the tide-generating force is concerned, distance is the more important variable than in case of the determination of gravitational attraction.

It is because of the greater distance between the sun and earth than that between the earth and moon that the moon dominates the tides.

Tide-generating force = m₁ m₂ / r3

The sun is 27 million times more massive than the moon. So the sun should have a tide- generating force 27 million times greater than the moon. But the sun is 390 times further away from the earth than the moon. Its tide-producing force, therefore, is 390³ times less than that of the moon. Thus the sun’s tide-producing force is about 46% or 27/59 that of the moon.

The tides generated in the oceans are the result of the rotation of the earth and moon about their common centre of mass which is about 4700 km away from the earth’s centre.

Actually this common center is located 300 m below the surface of the earth facing the moon. As the earth and the moon rotate as a system around this point (the common centre), all particles that make up the earth follow circles of equal radius.

If the earth is divided into a great number of equal mass, the centripetal acceleration required to keep each particle of the earth following an identical orbit is the same. The centripetal force that provides the acceleration comes from the gravitational attraction of the moon.

Though it is true that in order to keep the earth in its proper path the average gravitational attraction per unit mass must be equal to the average centripetal acceleration for different particles of the earth mass, the two, however, are not equal for all points of the earth.

It should be borne in mind that the centripetal acceleration for all particles is the same and directed towards the centre of each particle’s orbit. But the gravitational attraction of the moon that supplies this acceleration is greater for particles closer to the moon, and is directed towards the centre of the moon.

In order to understand the tide-producing forces, the influence of the moon should be considered first. Since the distance between the earth and moon is so large, the total of the attracting forces between all mass particles of the earth and of the moon reaches a complete equilibrium with the total of all centrifugal forces.

However, the centrifugal forces result from the motion of the earth and moon around their common center of gravity. But the equilibrium applies only for the total of the two forces, if the earth and moon are considered entities.

Since all points on the surface of the earth describe the same path during a revolution around the common earth- moon center of gravity, the centrifugal force on the earth has everywhere the same magnitude and direction.

But the forces of attraction have a different direction, and differ in magnitude depends on the distance of the points on the earth from the moon. In the attraction of the moon is maximum at the point Z which is closest to the moon and minimum at the point N on the earth’s surface.

The attracting force exceeds the centrifugal force on the hemisphere facing the moon. On the other hand, the centrifugal force is larger than the attracting force on the hemisphere facing away from the moon.

However, both of these forces cancel each other in the centre of the earth (E). At all other point’s small residual forces are acting as the resultant of these two forces. Actually these small residual forces act as the tide-generating forces.

Actually the tide-generating forces are very small in magnitude compared to the gravita­tional forces. The tide-generating forces may be separated into vertical and horizontal compo­nents.

Let us remember that the vertical compo­nent acts in the direction of the gravitational force. The vertical component is, therefore, of little importance in the generation of tides.

The vertical component, it may be kept in mind, has the same order of magnitude as other forces that act in the same direction in the ocean. The distribution of horizontal component of the tide-generating force over the entire globe.

The position of the tide-generating heavenly body (moon) at the equator has been shown in the zenith of Z. However, this figure represents a special case where the moon is supposed to be at the equator above point Z.

Like the moon, the sun is also the source of a tide-generating force, which is, however, not even half of the tide-producing force of the moon. As already stated, the tide- generating force is given by the difference between the force of mass attraction and the centrifugal force.

These forces depend on the intervening distance between the earth and the sun or the moon. The system of tide-generating forces, therefore, will differ in accordance with the changing distances between the earth and the heavenly bodies. These variations account for the inequalities in the tidal phenomena.

As a result of the rotation of the earth – moon system around the common center of gravity, the centrifugal force causes the earth and moon to have a tendency to fly away from each other.

However, this tendency is blocked, because the centrifugal force is exactly balanced by the gravitational attraction between the earth and the moon. Nevertheless, the centrifugal force causes the earth’s fluid hydrosphere to bulge out on the side opposite to the moon.

This effect is one of the external forces that raise the water surface enough to produce a high tide. The earth rotates through this bulge in the hydrosphere every day, so the high tide ideally should occur once in each 24 hours.

Since the strength of gravitational attraction between two objects tends to decrease with the distance between them, the gravitational force exerted by the moon on the earth is always stronger on the side of the earth closest to the moon.

So on that side of the earth, the moon’s gravitational attraction is stronger than the average, and also stronger than the centrifugal force derived from the rotation of the earth-moon system. The ocean waters on the side of the earth facing the moon, under the impact of the stronger gravitational force, bulge towards the moon.

This is called the direct tide. This bulge of water is directly opposite that produced by the centrifugal force. The earth rotates through this bulge produced by the centrifugal force once each day. Remember that this bulge of water is called the indirect tide.

Thus, there are two high tides daily. Since they are 180° of longitude indirect apart, they are separated by 12 hours. Between these two bulges the waters recede as they are pulled towards the areas of high tides. Accordingly, there is a low tide mid- way from each high tide.

Since the moon actually takes 24 hours and 50 minutes to complete one revolution around the earth, a high tide tends to occur once every 12 hours and 25 minutes. In this way, once every lunar day, two high tides and two low tides are produced.

As stated earlier, the period of time between two high tides is called the tidal interval and its length is 12 hours and 25 minutes. However, this tidal pattern may not be the same everywhere. But the most common tidal pattern is two high tides and two low tides in a day.

The reason for this lag of time by 50 minutes each day is simple to explain. Actually, the lunar day, the time that elapses between suc­cessive passages of the moon across the meridian, must be somewhat longer than the solar day of 24 hours. The lunar day is of 24 hours and 50 minutes.

Thus, this lags results from the fact that as the earth is making its rotation on its axis in 24 hours, the moon has moved 12.2° to the east. So the earth must rotate another 50 minutes to have the moon again on the meridian of the observer. This 12.2° of eastward rotation for the moon can be computed, knowing that it completes a 360° rotation is 29.53 day, by:

360729.53 days = 12.2° per day.