Validity of netwon's laws on the earth's surface

failexam

We all know that Newton's laws of motion are only valid in an inertial frame of reference.
Our earth is a rotating frame of reference, so Newton's laws of motion must not work anywhere on the surface of the planet. Do you agree?

Doc Al

To include the effects of the earth's rotation, one must modify Newton's laws by the addition of various 'fictitious' forces. (As with any non-inertial frame.) For many purposes, these effects can be neglected and you can treat the earth's surface as being approximately inertial.

D H

What's this inertial frame of reference thing to which you are alluding? Name one.

It's obviously not a frame fixed with respect to the rotating Earth; it's a rotating frame. A non-rotating frame with origin at the center of the Earth doesn't work either. The Earth as a whole is accelerating toward the Sun, the Moon, Jupiter, Venus, etc. How about the International Celestial Reference Frame, the ICRF? It's constructed based on physical measurements so as to be non-rotating, but those measurements aren't perfect. The ICRF almost certainly is rotating by some currently unobservable amount. The ICRF with its origin at the solar system barycenter is an accelerating reference frame; the solar system is orbiting the Milky Way. What about a frame with origin at the center of the Milky Way? The Milky Way is accelerating, too.

There is no such thing as a true inertial frame, at least not one that we can find. There are instead frames that are approximately inertial. The ICRF is approximately valid as an inertial frame out to Pluto and beyond. An Earth-centered inertial frame works quite nicely for many Earth-bound applications, but there are times when one needs to account for the frame's acceleration. Even an Earth-fixed frame works quite nicely as an inertial frame for applications in which velocities, relative displacements, and time intervals are small.

failexam

Thanks for the help!

Nobelian

Yes the frame of reference (earth) is spinning all the time, but we are also spinning with it, so we are always in the same position relative to the frame of reference, and this is why we don't see any change, as an observer. The earth is also rotating around the sun, but we are also rotating with it, so we also feel no forces acting upon us, even when there are forces. This is like when astronauts are in free-fall in space. They are not experiencing zero gravity. They are merely falling with their frame of reference, falling along the curve of the earth, similar to how we are falling along the curve of the sun, and along a curved circular path as we spin around on earth's axis. Newton's laws are valid within a frame of reference (relative to that frame of reference). Therefore they will not be affected if there is no change in our motion relative to our frame of reference, so I don't see why they wouldn't be valid on the earth's surface.

D H

When we look at a star, we see that star as undergoing uniform circular motion with an angular velocity of 360 degrees per day. If Newton's 1st and 2nd laws are valid, that uniform circular motion means there must exist some centripetal force that causes this observed motion. Suppose the star is a 1 solar mass star and is 10 light years away. That's a centripetal force of 10³⁹ newtons! Now compare to an observer on some non-rotating platform out in space. That person sees that same star as motionless. That means no net force. So which is right, a force of 10³⁹ newtons directed toward the Earth, or no force at all?

The answer is both if you create some fictitious forces to account for that perceived acceleration in the rotating frame.

The word is orbiting or revolving, not rotating. It's an important distinction.

BTW, we do feel a force. It's called the tide.

No, they aren't, at least not without fictitious forces. It's an accelerating frame.


Without the aid of these fictitious forces, Newton's 1st and 2nd laws are not valid in a rotating or accelerating frame. Now there's another problem: There is no equal but opposite force to these fictitious forces.

These fictitious forces can be a very useful fiction. For example, it would be downright impossible to simulate the weather from the perspective of an inertial frame.

Darwin123

Yes, of course. For observers fixed on the surface of the earth, Newton's Laws don't strictly apply. The two pseudoforces, centrifugal force and Coriolis force, don't satisfy the third law of motion.
Newton's Third Law of Motion (Principia, Law III) states:
To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and directed to contrary parts.
Read the third law carefully. According to the Third Law of motion, every force has to be associated with two bodies. If the bodies weren't paired, then the forces couldn't be paired.
Consider any body in the surface frame. Neither the centrifugal force nor the Coriolis force has a corresponding body to which a mutual action can be applied. From the standpoint of an inertial frame, the motions caused by these two pseudoforces are caused by a force on the observer, not on another body.
Violations in the Third Law are apparent when bodies are unpaired. If a force is exerted on one body and there is not corresponding body to which the opposite force can be applied, then the Third Law is violated.
As an example, consider the bob on a Foucault pendulum. At any point except on the equator, the Coriolis force makes the bob precess. However, there is no body corresponding to the bob upon which a mutual force can be applied.
In an inertial reference frame, there is no Coriolis force. The apparent motion of the Foucault pendulum doesn't occur. It is due to a force on the observer that is fixed to the surface of the earth.
For the POV observer in orbit around the earth, neither centrifugal force nor Coriolis force exists. There are only contact forces and gravitational forces. However, most humans observe from the POV of an local observer fixed to the surface of the earth. Therefore, the third law takes a slight beating in terms of local observation.

Darwin123

This is incorrect. Of course one could simulate the weather from the perspective of an inertial frame. One includes the contact force of the earths surface, the frictional force due to the atmosphere's viscosity, and the other "real forces". However, the acceleration of the earth's surface is not zero in the inertial frame.
In the end, one has to subtract the "background motion" of the earth's surface. However, none of this violates Newton's Laws.
In actual practice, the problem is simplified if the acceleration of the earth's surface is arbitrarily set to zero. No "background motion" has to be subtracted. The price of this simplification is that the Coriolis force and the centrifugal force has to be set to nonzero values. Thus, this is really a noninertial frame.
General relativity complicates this view of nature, somewhat. However, it ends up the same. The surface of the earth is not a geodesic. So Newton's Laws can't be valid even as an approximation from the standpoint of a surface frame.

D H

In theory, yes. In practice, no. Weather and climate simulations need to produce timely forecasts, the forecasts need to be reasonably accurate, and the programs must live within the constraints of a real computer. The final item is tough enough as is. Weather and climate programs can stress even the most powerful supercomputers. Your inertial frame weather simulation would violate at least one of these requirements. A non-rotating, Earth-centered frame is not inertial. It is an accelerating frame, and weather forecast programs do account for tidal forces. A weather forecast program that does its computations in the International Celestial Reference Frame is fundamentally broken. It won't satisfy any of those essential requirements.