PC1672 Advanced dynamics

3.12 Tides

Tidal forces arise from a combination of inertial forces and a nonuniform gravitational field.

Someone trapped in a lift that is falling freely in a completely uniform gravitational field would not be able to distinguish their situation from a lift at rest far from any gravitating object. This because the force of gravity on them is exactly cancelled by the inertial force that results from their use of a noninertial frame. The gravitational field of the Earth is not quite uniform: it always points towards the centre of the Earth and its strength increases as one gets closer to the centre. The inertial force on someone trapped in a lift falling to the Earth balances the force of gravity at their centre-of-mass, but gravity at their feet is stronger than the inertial force, while at their head it is weaker. As a result they feel a net force stretching them vertically. The radial form of the Earth's field means that there is also an unbalanced horizontal component of gravity on the person's hands and so the unfortunate person also feels a force squashing them sideways.

These tidal forces are are very small here on Earth, but they are much larger near denser objects. They are forces that will get you if you fall into a black hole (with the help of Andrew Hamilton).

Tidal forces also affect objects that are in orbit. For example, from the viewpoint of a frame fixed to the Earth, the gravitational force of the Sun is cancelled by the centrifugal force arising from the Earth's orbital motion. However this cancellation is exact only at the Earth's centre-of-mass. At the Earth's surface there is are residual tidal forces stretching the Earth along the Earth-Sun axis and squashing it sideways. This force creates two tidal bulges in the Earth's ocean, one pointing towards the Sun, the other away from it. The height of these bulges is about 0.24 m (in the absence of interactions with coasts and continental shelves which can distort them significantly).

The same thing happens as the Earth and Moon orbit their common centre-of-mass. The height of the tides raised by the Moon is about 0.56 m. This is larger than the tidal effect of the Sun even though the Sun's gravitational field at the Earth is much stronger. The Moon is so much closer to us than the Sun that the gradient of its field is larger.

Textbook references

K&K 8.4 examples 8.4, 8.5
B&O 8.2
M&T 5.5

Mike Birse
17th May 2000