The Moon's pull along the sea

Adapted from IOAA 2008 (Bandung), theory problem 8. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The pulls of the Sun and the Moon raise and lower the surface of the sea. Two points A and B lie on the equator, with a difference of longitude $\phi$ between them, and A is on the sea. The Moon is overhead at B. Find the horizontal acceleration of the sea water at A caused by the Moon, as measured by observers on Earth, in terms of $\phi$, Earth's radius $R$ and the distance $r$ from Earth's centre to the Moon.

AnswerThe horizontal acceleration towards B is $(GM\sin \phi /r^2)[(1+R^2/r^2-2(R/r)\cos \phi {)}^{-3/2}-1]$, where $M$ is the Moon's mass. It is close to $(3GMR/2r^3)\sin 2\phi$, largest 45° from the point under the Moon and zero under the Moon and 90° from it.