If we move the object, the force changes. The rate of change of the force is

The change in force, AF, in going from r to r + Ar is

Note that the tidal effects fall as 1/r3, faster than the 1/r2 fall-off of the gravitational force itself.

Example 23.3 Tidal effects on the Earth Compare the strength of the tidal effects exerted on the Earth by the Sun and the Moon.

Even though the Sun exerts a greater gravitational force on the Earth than the Moon, the closeness of the Moon makes its tidal effects greater.

The tidal effects of the Sun and Moon are responsible for the ocean tides on Earth (Fig. 23.25). We first look at how the Moon affects the Earth. We look at three points: (1) the point closest to the Moon, (2) the center of the Earth, and

(3) the point farthest from the Moon. We see that a1 > a2 > a3, as viewed from the rest frame of the

Moon. However, as viewed from the Earth, all accelerations must be relative to a2. In this frame the acceleration of (1) is a1 — a2 towards the Moon; that of (2) is zero; that of (3) is a3 — a2, and is therefore directed away from the Moon. This means that there will be a high tide on the side nearest the Moon and also on the side farthest from the Moon. We can think of the tide on the near side as the water on that side being pulled away from the Earth, and we can think of the tide on the far side as the Earth being pulled away from the water.

The Sun produces a similar effect, but only half as great in size. When the Moon and Sun pull along the same line, the difference between high and low tides is the greatest. When they pull at right angles to each other, the difference between high and low tides is the least. The tides on the oceans, where the water flows without obstruction, are well approximated by Fig. 23.25. However, in narrow or blocked waterways that run into the ocean, the flow of the water is quite complicated, and the time of high tide may be delayed by a few hours behind the time of the ocean high tide.

The height of the water tide is actually not found to be as large as one would calculate. This is because the Earth is not solid. It is therefore also distorted by these tidal forces. This constant

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