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# Orbits and Special Relativity watch

1. A clock is placed in a satellite that orbits the earth with a period of 90 min. By what time interval will this clock differ from an identical clock on the earth after 1 year?

My thoughts:

Use

Where Te is the period of the satellite as viewed in the reference frame of earth (i.e. due to time dilation) and Ts is the period of the satellite in its reference frame.

First find v using where in this case and R is the orbital radius.

Found Orbital Radius using Kepler's laws, and after a bit of algebra, I got v = 7751 m/s = (2.5E-5)c m/s.

Use the standard formula for the factor ... but this gives me , unsurprisingly. So, obviously, ...

Have I done this wrong?
2. (Original post by trm90)
A clock is placed in a satellite that orbits the earth with a period of 90 min. By what time interval will this clock differ from an identical clock on the earth after 1 year?

My thoughts:

Use

Where Te is the period of the satellite as viewed in the reference frame of earth (i.e. due to time dilation) and Ts is the period of the satellite in its reference frame.

First find v using where in this case and R is the orbital radius.

Found Orbital Radius using Kepler's laws, and after a bit of algebra, I got v = 7751 m/s = (2.5E-5)c m/s.

Use the standard formula for the factor ... but this gives me , unsurprisingly. So, obviously, ...

Have I done this wrong?
Probably not, use a binomial expansion for gamma.
3. (Original post by TableChair)
Probably not, use a binomial expansion for gamma.
Ah okay... so I used and I managed to get 1 again, according to my calculator. Argh! The higher order terms probably wont change anything cause they're too negligible...
4. (Original post by trm90)
Ah okay... so I used and I managed to get 1 again, according to my calculator. Argh! The higher order terms probably wont change anything cause they're too negligible...
You're putting in numbers too early. Never put numbers into an equation until the end.

Find an algebraic equation for the difference in time.

And yes, the higher order terms are negligible, hence why you should be using a binomial expansion.
5. (Original post by TableChair)
You're putting in numbers too early. Never put numbers into an equation until the end.

Find an algebraic equation for the difference in time.

And yes, the higher order terms are negligible, hence why you should be using a binomial expansion.
Right, so, from the start I guess (excluding velocity part)

The time difference is:

. Using lower case t here to distinguish between time and period, a mistake I made before I believe.

But

And now when I plug in v, I get gamma -1 = 3.125E-10, and minutes (after I converted a year into minutes).

That works, right?
6. (Original post by trm90)
Right, so, from the start I guess (excluding velocity part)

The time difference is:

. Using lower case t here to distinguish between time and period, a mistake I made before I believe.

But

And now when I plug in v, I get gamma -1 = 3.125E-10, and minutes (after I converted a year into minutes).

That works, right?
Seems fine, but you might want to use seconds instead of minutes
7. (Original post by TableChair)
Seems fine, but you might want to use seconds instead of minutes
Cool - thanks!

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