Orbital ForcesThe dominant effect of gravity is first and foremost the orbital forces that keep planets and their moons in their relative orbits around each other and around the Sun. The forces experienced by planet 1 and planet 2 are the same but their accelerations are in inverse proportion to their masses.
FOrbital = m1a1 = m2a2 = GNm1m2/(d1-2)2GN is Newton's gravitational constant (big G), and d1-2 is the distance between the centers of planet 1 and 2.
WeightAnother common effect of gravity is the phenomenon of weight. Weight is a force arising from the acceleration towards the center experienced by any mass (m) at the surface of the Earth. It is commonly denoted g (small g).
FWeight = mg = mGNmEarth/rEarth2Here rEarth is the radius of the Earth.
FTide = matide = mGNmMoon((1/dEarthCenter-Moon)2 - (1/dEarthSurface-Moon)2)To first approximation, this is at most
FTide = matide = 2mGNmMoonrEarth/(dEarth-Moon)3
Taking into account the finite radius of the Earth, rEarth, the tidal force is effectively an inverse cube law. The overall factor of 2 is another measurable reminder of the original inverse square law. We see that in magnitude,
FTide/FWeight = 2mMoon/mEarth(rEarth/dEarth-Moon)3 = 1.12×10-7Similarly, for the tidal force due to the Sun
FSunTide/FMoonTide = mSun/mMoon(rEarth-Moon/dEarth-Sun)3 = 0.455
On those rare occasions where the Moon and Sun tidal accelerations are aligned
FTide/FWeight = 1.12×10-7(1 + 0.455) = 1.63×10-7
Tidal forces on Earth are very, very small. Moreover they vary over time depending on the position of the Moon and the Sun. Today we can calculate the position of the Moon and the Sun with great accuracy, but historically, calculating the Moon's position from first principles was a difficult challenge. More than 60 years elapsed between Newton's Principia (1687) and Clairaut's Theorie de la Lune (1752).
It is important to realize that the Earth does not take tidal forces passively. In addition to ocean tides, atmospheric and earth tides are observed. The Earth strains in response to tidal forces and the actual effects measured at any given point are a combination of these forces and the Earth's response. Even though the driving tidal forces can be calculated accurately, the combined result is hard to predict accurately, to this day.