How Large Must a Spark Chamber Be to Qualify as an Inertial Frame?

[Ed Quanta]

I am reading Taylor and Wheeler's Spacetime Physics. I am enjoying it a lot and find it extremely readable but I have a question regarding something. I am not sure how to determine the dimensions of a frame necessary for it to be called a free float frame. This is a general question I know.

To be more specific, one problem I am working on discusses an earthbound laboratory in which an elementary particle passes from side to side, traveling at .96c through a cubical spark chamber one meter wide. The first question asks for what length of laboratory time is this particle in transit through the spark chamber.

This is 1.04 meters or 3.4 X10^-9 s of time. No problem.

Then they ask how far a separate test particle released from rest will fall in this time?

This is 6 x 10^-17 m. Once again no problem.

Now I understand that in order for the frame to be considered a free float or inertial frame, there must be no observable relative acceleration between the two particles.

They ask how wide the spark chamber can be and still be considered a float frame?

I am not sure.

Then they say to assume that the optical equipment being used can detect a test particle change of position as small as 5 * 10^-7 m.

Now I know that it takes the test particle being dropped from rest 3 X 10^-4 s to cover this distance. And the particle moving at .96 c, moves a distance of about 8.64 X 10^4 m in that time.

How now can I determine how long an earthbound spark chamber must be to be considered free-float for this sensitivity of detection?

Do I just find a time interval short enough that difference in position between the fast moving particle and particle being dropped from rest is less than 5 * 10^-7m?

 

[Almsco]

That seems too simple to be a proper answer.

It sounds like you are looking for some help understanding the concept of an inertial frame. A free float frame is an inertial frame, which is a coordinate system in which Newton's first law holds: an object at rest or moving at a constant velocity remains so unless acted upon by an external force. In order to determine the dimensions of a frame necessary for it to be called a free float frame, you need to calculate the relative acceleration between the two particles, taking into account their initial velocities. To do this, you need to find the time difference between the two particles and then the distance difference between them. If the distance difference is greater than 5*10^-7m, then the frame cannot be considered a free float frame.

 

[Entropia]

Or is there a more precise way to calculate the size of a free float frame?The size of an inertial frame is not determined by its physical dimensions, but rather by its properties and behavior. An inertial frame is one in which objects at rest or in uniform motion will remain in that state unless acted upon by an external force. In other words, an inertial frame is one in which the laws of physics, specifically the laws of motion, hold true.

In the context of the problem you are working on, the size of the spark chamber does not determine whether it is an inertial frame or not. Instead, it is the behavior of the particles within the chamber that determines this. As long as there is no observable relative acceleration between the two particles, the spark chamber can be considered an inertial frame.

To determine the size of the spark chamber in this scenario, you can use the given information about the sensitivity of the optical equipment and the time it takes for the test particle to cover a certain distance. You are correct in thinking that you need to find a time interval short enough that the difference in position between the fast-moving particle and the particle being dropped from rest is less than the sensitivity of the equipment (5 * 10^-7 m). This will ensure that the relative acceleration between the two particles is negligible and the spark chamber can be considered an inertial frame.

In summary, the size of an inertial frame is not determined by its physical dimensions, but rather by the behavior of objects within it. In this case, you can use the given information about the sensitivity of the equipment and the time it takes for the test particle to cover a certain distance to determine the size of the spark chamber that can be considered an inertial frame.