How to check if frame is inertial?

[hackhard]

inertial frame is one in which isolated particle has constant velocity
but is there actually any "isolated particle " ?
how then can frame be defined as or not being inertial ?
or is it that -
for a system in which acceleration due to external forces is equal for all members ,
the frame of member whose acceleration due to internal forces is negligible
(compared to acceleration due to internal forces of other members) is considered inertial for analyzing motion of other members of system

 

[Dale]

but is there actually any "isolated particle " ?

Attach an accelerometer to the particle. If the accelerometer reads 0 then the particle is "isolated" in the necessary sense.

 

[FactChecker]

The real problem with using an accelerometer would be in distinguishing between acceleration and gravity.

 

[bcrowell]

There are two different definitions of an inertial frame, one in Newtonian physics and one in relativity.

but is there actually any "isolated particle " ?

Of the four forces of nature, three can be shielded against, so a particle can be isolated from them in a well-defined sense. However, we don't have gravitational shielding, and the equivalence principle guarantees that gravitational shielding can't exist.

In Newtonian physics, we assume that we have an omniscient observer who knows where every mass in the universe is. We can then determine all gravitational forces and infer how a test particle would have moved if it hadn't been affected by gravity.

In relativity, we use the definition given in Dale's #2.

According to the Newtonian definition, the Earth's surface is inertial, and the ISS is noninertial. According to the relativistic definition it's the other way around.

There is a discussion of this sort of thing in section 8.2 of my book Relativity for Poets http://lightandmatter.com/poets/ , which is free.

 

[hackhard]

According to the Newtonian definition, the Earth's surface is inertial,

why is the Earth frame inertial if it revolves around the sun?
and how is the sun frame inertial if revolves about center of milky way?

 

[Drakkith]

why is the Earth frame inertial if it revolves around the sun?
and how is the sun frame inertial if revolves about center of milky way?

It's not an inertial frame. But we approximate it as an inertial frame most of the time unless accuracy requires us to do otherwise.

 

[A.T.]

why is the Earth frame inertial if it revolves around the sun?
and how is the sun frame inertial if revolves about center of milky way?

Think of a non rotating isolated planet. The surface is inertial in Newtonian mechanics, but non-inertial in GR. GR goes by what an accelerometer measures.

 

[hackhard]

But we approximate it as an inertial frame most of the time unless accuracy requires us to do otherwise

but we can approximate it as inertial frame only for earthly bodies

 

[CrazyNinja]

but we can approximate it as inertial frame only for earthly bodies

True. If as @Drakkith says in #6, the Earth being inertial is inaccurate for whatever it is that you are measuring, then take another body as an inertial frame. This body could be such that it plays a role similar to that which Earth plays for what you called "earthly bodies".

 

[hackhard]

IF

earth being inertial is inaccurate for whatever it is that you are measuring

how will i know if my measurements are inacurate ?

 

[Dale]

how will i know if my measurements are inacurate ?

This is why I use relativity's definition of inertial, which I described above.

I would highly recommend reading bcrowell's link for details.

 

[FactChecker]

how will i know if my measurements are inacurate ?

The application will determine how much accuracy you need. Here are a few assumptions that I would make:
1) If you are looking at the motion of things leaving the Earth, you definitely need the full Earth rotation and motion in the Solar System.
2) If you are looking at motion of things in orbit, you definitely need the full Earth rotation and probably can ignore motion in the Solar System.
3) If you are looking at long distance flights on Earth, you definitely need to consider the rotation of the Earth but can ignore the motion of the Earth in the Solar System
4) If you are only interested in short distance flights on Earth, you can probably assume that the Earth is not rotating or moving.
5) If you need incredible accuracy, you may need to consider more than indicated above.

Of course, there are always exceptions, but they are rare.

 

[David Lewis]

If a significant gravitational field is present, the frame is non-inertial.

 

[hackhard]

If a significant gravitational field is present, the frame is non-inertial.

again , "significant" in comparison to what?

 

[David Lewis]

Significant enough that you must take it into account to obtain your desired level of precision.

 

[FactChecker]

Significant enough that you must take it into account to obtain your desired level of precision.

For practical use, that is too strict. Most physics is done assuming that gravity is a force and does not prevent a coordinate system from being inertial. All the physics done before Einstein was like that. Until a person is dealing with relativity, he should do the same.

 

[David Lewis]

You're correct. There are alternate definitions for the term non-inertial that, while they may obscure what is really going on, make problem analysis simpler.

 

[hackhard]

so a frame can be taken to be inertial if pseodo forces in that frame are negligible compared to real forces
so centripetal acceleration of Earth due to sun = 0.0059 m/s^2
so Earth frame is inertial for motion of animals-- (pseudo force=40 kg * 0.0059 m/s^2)
so Earth frame is not inertial for motion of sun-- (pseudo force= 2 × 10^30 kg * 0.0059 m/s^2)

 

[FactChecker]

so a frame can be taken to be inertial if pseodo forces in that frame are negligible compared to real forces

I agree

so centripetal acceleration of Earth due to sun = 0.0059 m/s^2

Yes.

so Earth frame is inertial for motion of animals-- (pseudo force=40 kg * 0.0059 m/s^2)

Is that right? I don't know where that is coming from. It looks like it is derived from the acceleration of the Earth's orbit around the Sun. That doesn't seem right to me. It should be related to the Earth's rotation rate of 360 deg/day and it would be different at the Equator versus the North Pole.

so Earth frame is not inertial for motion of sun-- (pseudo force= 2 × 10^30 kg * 0.0059 m/s^2)

What do you mean?

 

[hackhard]

AT north pole ,earth frame is inertial for motion of animals-- (pseudo force on animal=40 kg * 0.0059 m/s^2)
thus pseudo forces need not be added for laws of motion to be valid on animal( from Earth frame )(AT north pole)

AT north pole , Earth frame is not inertial for analyzing motion of sun-- (pseudo force on sun= 2 × 10^30 kg * 0.0059 m/s^2)
thus pseudo forces must be added for laws of motion to be valid on sun( from Earth frame) (AT north pole)

 

[vanhees71]

why is the Earth frame inertial if it revolves around the sun?
and how is the sun frame inertial if revolves about center of milky way?

Indeed, the Earth is also spinning, and it's not an inertial frame. You can prove this with help of a pendulum. This famous Foucault experiment proves the rotation of the Earth around its axis. You find Foucault pendulae in many science museums and often in physics departments at universities :-).

 

[FactChecker]

AT north pole ,earth frame is inertial for motion of animals-- (pseudo force on animal=40 kg * 0.0059 m/s^2)
thus pseudo forces need not be added for laws of motion to be valid on animal( from Earth frame )(AT north pole)

AT north pole , Earth frame is not inertial for analyzing motion of sun-- (pseudo force on sun= 2 × 10^30 kg * 0.0059 m/s^2)
thus pseudo forces must be added for laws of motion to be valid on sun( from Earth frame) (AT north pole)

When you say it is inertial for motion of animals, I can't figure out if you are suggest ignoring the orbit of the Earth around the sun or the rotation of the Earth. To avoid confusion, I think we need to be precise about which effect is being talked about.

 

[hackhard]

i suggest ignoring orbit of Earth around sun
even then Earth frame must be non-inertial (for animal) but at poles (due to rotation about its own axis)
still Earth frame is taken inertial (for an animal at equator) even at equator because when life starts it has tangential speed of Earth surface and necessary centripetal force due to Earth keeps it at rest from Earth frame. thus centrifugal force always cancels centripetal force so none are taken into account.
thus Earth frame is non-inertial at equator for meteor striking earth

 

[hackhard]

Attach an accelerometer to the particle. If the accelerometer reads 0 then the particle is "isolated" in the necessary sense.

but accelerometer measures gforce. if accelerometer reads 0 , gforce is 0.
but in freefall accelerometer reads 0 but body not isolated?

 

[Dale]

but accelerometer measures gforce. if accelerometer reads 0 , gforce is 0.
but in freefall accelerometer reads 0 but body not isolated?

Since a body cannot be isolated gravitationally (see bcrowell's post #4) the modern definition of "inertial" does not require isolation. The modern definition is much simpler and more practical: if an accelerometer reads 0 then it is inertial. An object in free fall is inertial by this definition.

 

[vanhees71]

In other words, a local rotation-free reference frame at rest relative to a freely falling body defines a local inertial frame. That't the (weak) equivalence principle and starting point for General Relativity. The emphasis is on the word local!

That also explains why the motion of the Earth around the Sun is not an issue (locally): It's very close to free fall and thus defines an inertial frame, but the Earth is also rotating around its own axis and that's why a reference frame at rest relative to a point on the Earth's surface is only approximately an inertial frame. The angular velocity of the rotation is very small, i.e., ##2 \pi/1 \text{day}## compared to typical motion of objects close to Earth like a free falling stone. In principle the stone has a deflection in eastern and southern direction, but it's very hard to measure. The Foucault pendulum is an exception, because you observe it over a long period, and thus you observe the effect of the Coriolios inertial force (1st order in ##\omega##) in the rotation of the pendulum's plane of motion.

 

[hackhard]

,but in freely falling(due to Earth gravity) frame (ie inertial frame) no inertial forces are added.
then why a nearby free falling object has 0 acceleration in that frame

or will pseudo force be added because its local inertial frame ?

 

[Dale]

,but in freely falling(due to Earth gravity) frame (ie inertial frame) no inertial forces are added.
then why a nearby free falling object has 0 acceleration in that frame

or will pseudo force be added because its local inertial frame ?

In this approach Gravity is not a force, it is geometry. So a nearby free falling object is not subject to any force (no gravity and no fictitious force) and therefore travels inertially.

Here is some more details
https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/