I have a question about tidal forces on the far side of a body experiencing gravitational attraction from another body.
Let's assume we have two spherical bodies $A$ and $B$ whose centers are $D$ apart, and who have radiuses $R_{A}$ and $R_{B}$, much smaller than $D$.
Gravitation force has a law in $1/distance^{2}$. On the line $AB$, a mass $m$ at the point point of $B$ closest to $A$ experiences a pull towards the center of $A$ of magnitude $Km/(R_{B}-D)^{2}$, and the point farthest from $A$ experiences a pull of magnitude $Km/(R_{B}+D)^{2}$.
Note that I did not mention the direction of the force at that second point. It seems that the centers of mass being all on one side of that point, the force should point towards the center of $A$ on the line $AB$. However, the tidal bulge on this "far side" suggests that some force (?) is pulling on matter away from the center $A$.
How can we explain this tidal bulge on the far side? – I am specifically interested in a clear derivation using classical mechanics.
Best Answer
It has to do with the fact that the entire earth is in an accelerating reference from due to the attraction from the moon.
Imagine 3 points on a line out from the moon. $N$ is the oceans near the moon, $C$ is the center of the earth, and $F$ is the far oceans. The attraction falls off with distance so the forces $F_N > F_C > F_F$ as you know. Also, all the forces are directed towards the body $A$ as you know.
Since the oceans are not rigidly attached to the earth, these points all can move independently to some degree. Due to the high force, $N$ will get pulled towards $A$ by a large amount. $C$ will get pulled a little. While $F$ will barely get pulled at all. What's important is that due to the different amounts they move, the distances between the three points have increased. $N$ and $C$ are farther apart than they were before, and $C$ and $F$ are farther apart too.
Now imagine your standing on the earth at $C$. The oceans at $N$ are farther away from you now, so the ocean appears to bulge out at that point. This seems to make sense since that side is closer to the moon. But also, the oceans at $F$ are farther away from you too, so they also appear to be bulging.
In essence, it is not that those far oceans are being pulled away from the earth, it's that the earth is getting pulled away from those oceans.