A simple relativistic time question

[Dodsy]

Before continuing, I wanted to verify I have fixed the issue with the wording of my variables, and, I have seen answers for this question elsewhere. I also know I'll get full marks regardless of how I use my equations, as this question obviously isn't very focused on that, however, for my own purposes I would appreciate a little guidance. I feel that everywhere I look people are doing this wrong. I want to make sure I'm at least doing it right this way.

Thanks. Dodsy.


1. Homework Statement

Homework Equations

The Attempt at a Solution

I figure, the time of the clock in orbit will run SLOWER than the clock not in orbit. So, I switched the equation around a bit. First, let's get our variables in order here, and add some fake values for things.

A)

Some proofs:

B)

Some proofs:

I guess what I'm asking is, am I using the right equation? It's only logical to assume that the Earth's representation of a year's time in seconds would be the Tm and the orbit's representation of a year's time in seconds would be the Ts. Any input?

 

[MachPrincipe]

Before continuing, I wanted to verify I have fixed the issue with the wording of my variables, and, I have seen answers for this question elsewhere. I also know I'll get full marks regardless of how I use my equations, as this question obviously isn't very focused on that, however, for my own purposes I would appreciate a little guidance. I feel that everywhere I look people are doing this wrong. I want to make sure I'm at least doing it right this way.

Thanks. Dodsy.1. Homework Statement
View attachment 92623
View attachment 92624

Homework Equations

View attachment 92625[/B]

The Attempt at a Solution

I figure, the time of the clock in orbit will run SLOWER than the clock not in orbit. So, I switched the equation around a bit. First, let's get our variables in order here, and add some fake values for things.

View attachment 92626

A)
View attachment 92628

Some proofs:
View attachment 92630

B)
View attachment 92632
Some proofs:
View attachment 92633I guess what I'm asking is, am I using the right equation? It's only logical to assume that the Earth's representation of a year's time in seconds would be the Tm and the orbit's representation of a year's time in seconds would be the Ts. Any input?

Assuming that the gravitational field is not strong - Earth - and the speed of the satellite is not relativistic - Eart orbit - you can approcimaxe the result by
delta t' = delta t * sqrt ( 1 -v^2/c^2 + delta phi/2/c^2)

where phi es Newtonian gravitational potential.
You must take into account the potentisl at the lab and also the rotation of Earth, using twice the above formula, one for Earth and one foor satellite, being the reference point the center of Earth and a non. rotating inertial system placed there.

 

[Dodsy]

That is a completely logical thing to assume, and I fully understand why you would think that. But it's beyond the scope of the course which is introductory in nature. For instance, on time dilation the text merely demonstrates that objects moving at a higher velocity will experience the dilation. Then the lesson is over, and the next lesson talks about length contraction, mass, and energy equivalence. But never does it mention gravity having an effect on the clock. I believe we are to assume that the clock on Earth will run at 3.154 x 10⁷s.

I'll add into my answer that my equations assume that the clock on Earth measures time to be 3.154 x 10⁷s.

Thanks, and once again you are not incorrect at all.

 

[gneill]

I presume that for this exercise you are meant to account for time differences due to velocity differences alone and ignore other factors such as gravitational potential which also has some effect on relative rates of time.

Although arbitrary your satellite velocity looks a bit strange. First, a squared velocity should have units of $m^2/s^2$, not $m/s$. Second, I would expect the square of the velocity for an Earth satellite have an order of magnitude closer to $10^6$ or $10^7$ $m^2/s^2$. Also you stated that the speed of light squared is 3.0 x 10⁸ m/s, which it is not. That's just the speed of light.

Rather than try to work with numbers that are arbitrary anyway, why don't you work with the ratio $\beta = v/c$? It's a well known convention. Then you can say something like, "Suppose that the satellite's speed in orbit is 1/100,000 c...", and then for the case where the speed of light is presumed to be much lower the value of $\beta$ becomes 1/2.

Using $\beta$ your dilation factor becomes $\gamma = \frac{1}{\sqrt{1 - \beta^2}}$.

Then you can say that when a year has passed on the Earth bound clock, the satellite borne clock will read only $year/\gamma$. You can work out how to find the difference in seconds between the readings if you need to.

 

[MachPrincipe]

That is a completely logical thing to assume, and I fully understand why you would think that. But it's beyond the scope of the course which is introductory in nature. For instance, on time dilation the text merely demonstrates that objects moving at a higher velocity will experience the dilation. Then the lesson is over, and the next lesson talks about length contraction, mass, and energy equivalence. But never does it mention gravity having an effect on the clock. I believe we are to assume that the clock on Earth will run at 3.154 x 10⁷s.

I'll add into my answer that my equations assume that the clock on Earth measures time to be 3.154 x 10⁷s.

Thanks, and once again you are not incorrect at all.

Then get the speed with v = sqrt(2*G*M / r) and apply it to usual time dilation formula. You need to calculate Earh rotation effect on the lab clock, still, due to speed.

 

[MachPrincipe]

BTW, if I am allowed to ask it, what kind of introductory course is this? High school? Because of it is University degree, I cannot believe they can mix concepts so wrongly as to confuse the alumni in a later stage so badly.

 

[Dodsy]

Yes this is just high school, and has just been placed in the incorrect area of the forum by another moderator. Gneil, sorry about the squared variables, I made a disclaimer at the beginning of my post, as I've currently changed this in my variables. Thanks for you help guys, I appreciate it, I'll look over the issues a little bit more.

Also, Gneil, Would you mind moving this back to introductory physics homework?

 

[MachPrincipe]

Yes this is just high school, and has just been placed in the incorrect area of the forum by another moderator. Gneil, sorry about the squared variables, I made a disclaimer at the beginning of my post, as I've currently changed this in my variables. Thanks for you help guys, I appreciate it, I'll look over the issues a little bit more.

Also, Gneil, Would you mind moving this back to introductory physics homework?

Well, you are luck to be given an introduction to relativity. When I studied high school, only classical physics was reviewed - no relativity, no quantum physics or mechanics. But well, the (man-made) laser light was not invented yet! :)

 

[Dodsy]

Yes, there is an introduction to both relativity and quantum mechanics, as well as an assessment of the implications of each, which goes further in depth as to how to use the equations given when dealing with length contraction, mass, or energy equivalence.This is not advanced physics to say the least.