An Idiot’s Guide to: The Tides and Tidal Forces
What are Tidal Forces?
An Idiot’s Guide to: The Tides and Tidal Forces It is a common misconception among the public that gravity is just a force going down. Of course, with Newton’s theory of universal gravitation, this simply isn’t so. Rather, gravity is in truth an all-present fundamental force in this universe, pulling all massive objects towards each other. Another misconception is the so-called uniform gravitational field. While, given the large size of the earth’s radius in comparison to any movement you might realistically encounter terrestrially, the effect is usually small enough to be ignored, the gravitational field is anything but uniform. Like many forces, the strength of the attraction is proportional to inverse square of the distance. So as you get further away, the gravitational force gradually reduces, but never quite goes away. Newton summarised all this in the equation:
Where G is the gravitational constant, M or m are the masses of the two objects that are being regarded and R is the distance between the two centres of masses. So how does this lead to Tidal Forces? The important point is that the gravitation force attracting each part of one object to the other object varies greatly according to the distance between them. When you double the distance, you have only a quarter of the force.
Consider the diagram below: (Note: Nowhere near to scale)
In this diagram, it can be clearly seen that Point A on the Earth is closer to the Moon and its centre of mass than point B, or point C. Hence, with the formula above, it is more strongly attracted than B or C. As a result, A would be pulled towards the moon more than B, and even more than C. This difference in attraction is the fundamental factor in tidal forces, as it makes the object be stretched in the direction of the force. Of course, this factor would be much greater closer to a massive object, where the overall forces involved and the relative differences in distance would be greater. This force is proportional to M/r^3.
The effect is more interesting when we have an object rotating around another.
In the diagram above, it is obvious that the linear velocity of A>C>B, since the three all have the same position in the orbit.
However, in a circular orbit, we have the formula: v^2 = GM/R
Where R is the radius of the orbit, so the linear velocity of orbiting objects should decrease with increasing distance. In short, while C has the right velocity for the orbit, A is travelling slightly too fast and B is travelling slightly too slowly. Thus, while A tends to try to go to a higher orbit, B tends to enter a lower orbit, pulling the object apart. The only reason objects like planets don’t fly apart into a broad disk around the centre of their orbits is because of internal gravitational forces that hold them together.
Real World Examples and Tidal Braking
An example of the first sort of tidal force is a simple hypothetical journey into a black hole. As we enter the black hole, an astronaut would quickly come under severe tidal forces as he gets closer and closer to the singularity, the centre of the black hole’s mass. The force pulling on his feet (assuming he went in feet first) would quickly become much greater than that on his body and much much greater than on his head. Soon he will become stretched out. Considering the strength of the black hole’s gravity and that the average human can only withstand 12 g difference between feet and head, he would very quickly be torn apart and killed, in a process dubbed spaghettification. Paradoxically, the smaller the black hole, the stronger the tidal forces. This is because the distance to the singularity is much smaller, and so the relative difference in gravitational acceleration is lower despite the increased mass.
A more complex example is if we consider the earth’s tides. In this, case, not only do we have the Earth’s orbit about the sun (solar tides) but also the moon’s orbit about the earth, providing the so called lunar and solar tides. The fact the tidal forces considers the relative distances from the moon and the relative differences in force caused by this is why we have two tides a day. Though apparently these tides are exhibited solely by the sea and oceans, in fact the entire earth is deformed by the tidal forces. Every 12 hours, the entire earth’s diameter shifts by up to a foot (0.3 m) due to the moon’s gravitational tidal force. The fact we see tides as a purely water based phenomenon is because water is that much less rigid than the rock, and we only see the difference between the moving earth and the moving water.
But because the rotation of the earth is much faster than the rotation of the moon around it (periods being a day as opposed to a month), the peak of the tides move around the earth, drawing water with it around the world. This vast movement of water causes friction with the sea floor etc, which works to reduce Earth’s rotation. However, a basic law of classical physics is the conservation of angular momentum. Thus, the actual effect of the tides is to slow down the earth’s rotation and speed up the moons, moving it further and further away, until the period of the earth’s rotation is the same as that of the moons, and we have NO lunar tides. This effect is called “tidal braking”.
What are the implications of this? We have actually estimated the work done on the sea floor on the tides and have in fact calculated the rate of change of the rotation of the earth as 2 hundred millionths of a second per rotation. That goes to 14 seconds per century. If we extrapolate backwards into time, we can apparently see a time where the earth and the moon are in the same place, spawn many theories that the moon was formed from the earth. If we extrapolate forward into the future, we see a point where the moon is so far away we can barely see it, and the earth is tidally locked with the rotation of the moon. But remember, the solar tides will still continue to slow Earth’s rotation with the solar tides, so that any lunar tide would work to speed up Earth instead of slowing it down… But these timescales are way way ahead, probably beyond the lifetime of the earth itself. So don’t worry, the tides will be with us for a long time.