Why is angular momentum conserved?

My intuition is if an object is orbiting a centre, it is accelerating as the direction of its vector constantly changes, i.e a ball orbiting a stick because they are tied by a string. I dont understand why Earth's spin does not slow down, if we think of Earth as lots of individual atoms, those atoms are all orbiting the earth's axis without slowing down yet their direction is always changing. It is as if a force is applied to them to change their direction. Please guide/correct me.

BvU
Homework Helper
a force is applied to them to change their direction
There is. Gravity holds things together. But that force does no work in the direction of motion.

CWatters and Jonathan1218
yea... ty

Ibix
2020 Award
If you think about the speed of each atom of the earth, it doesn't change. Only the direction changes - and energy doesn't care about the direction, only the square of the speed. So the energy doesn't change. So why would the earth slow down?

Jonathan1218
LURCH
For this type of question it might be counter productive to keep using the model of gravity in which the Earth going around the sun is compared to a ball swinging around on a string. Gravity is actually a curvature of space-time (a somewhat difficult concept), which a rather different situation. In orbiting the Sun, the Earth is actually following a “straight line at a constant speed”, which the same as setting still.

Unless of course we include the effects of gravity waves. They do bleed off some of the planet’s orbital velocity. But in the case of the Earth-Sun system, this effect would be very small.

BvU
Homework Helper
Beg to differ. There are relativistic effects, to be sure, but they come at a level of detail that can better be postponed until a number of non-relativistic details have been worked out using Newtonian gravity. The earth and moon orbit around a common center of mass (which is why we have high and low tides twice per day instead on once). The moon is already in a tidal lock and the earth is going that way (slowly!) -- so the earth spin is in fact slowing down ! And the moon moves away from us !

russ_watters and Ibix
PeroK
Homework Helper
Gold Member
2020 Award
For this type of question it might be counter productive to keep using the model of gravity in which the Earth going around the sun is compared to a ball swinging around on a string. Gravity is actually a curvature of space-time (a somewhat difficult concept), which a rather different situation. In orbiting the Sun, the Earth is actually following a “straight line at a constant speed”, which the same as setting still.

Unless of course we include the effects of gravity waves. They do bleed off some of the planet’s orbital velocity. But in the case of the Earth-Sun system, this effect would be very small.

This is a "B" level thread!

russ_watters and BvU
Ibix
2020 Award
I think gravity is an irrelevance here anyway - OP seems to me to be talking about the Earth's daily revolution about its axis, not its orbit around the Sun.

Basically the reason rotating objects do slow down is friction. If you don't have that (e.g. a planet spinning in a vacuum) then the reason they don't slow down is that nothing really changes. Take a snapshot of a spinning planet from above the pole, and then another. Rotate the camera when you take the second picture and you won't be able to tell it apart from the first.

There is. Gravity holds things together. But that force does no work in the direction of motion.

I beg to differ. Every atmospheric molecule with a downward component of motion has the speed of that motion accelerated by gravity. Every atmospheric molecule with an upward component of motion has the speed of that motion decelerated by gravity. That is why a mass of air moving downward increases in temperature and an upward moving mass decreases in temperature.

I dont understand why Earth's spin does not slow down, .

It does slow down. The mean length of the daylight period is diminishing at the rate of 2.3 milliseconds per century.

russ_watters
Mentor
I beg to differ. Every atmospheric molecule with a downward component of motion has the speed of that motion accelerated by gravity. Every atmospheric molecule with an upward component of motion has the speed of that motion decelerated by gravity.
I'm sure @BvU knows that, but in the scenario being discussed - the rotation of the Earth about its axis - the acceleration is perpendicular to the direction of motion.

LURCH
I see that I mistook the OP for a question about the Earth’s orbit, when it was actually about revolution on the axis. However, I believe what he was asking (Janathan, correct me if I’m wrong here) is basically,”how is this NOT perpetual motion?” Although it is true that tidal forces from the Moon are gradually slowing the planet’s rotation, I don’t think that’s what is being asked. If the Moon were not slowing the Earths rotation, would it keep spinning forever? If so, how does one account for the acceleration that keeps each part turning round? If not, what would slow it down?

That is what you were asking, yes?

bobob
Gold Member
The answer to your question is both simple and difficult to explain in detail without knowing your background. The simple part is that one can show in general that invariance under some continuous symmetry automagically gives you a conservation law. This is called Noether's theorem. If you make the assumption that the laws of physics do not change if you face a different direction. i.e., there is no preferred direction in space singled out by nature, then you get conservation of angular momentum as a result. I am answering the question I think you actually asked, which was about the conservation law.

CWatters