S DILEMMA: Time Dilation in Simple Harmonic Motion?

[THE HARLEQUIN]

i have been wondering about this for a while ...

suppose , a spacecraft in space moves with a simple harmonic motion ( e.g. on a straight line ) between points on space A and B . it fires a rocket from point B that travels with a speed .9c ... now
if the rocket comes back to point B after a time t ( measured in the spacecraft ) then what amount of time will be counted by the person driving the rocket ?
2. what will be the measured time if the rocket comes back and land on any points along the line AB ?
[ the spacecraft always has the same frequency all the time ]
3. what are the graphical representation of the above situations ?

thanks for your answers in advance ..
THE HARLEQUIN

 

[Dale]

I don't understand your set up at all. Can you write down algebraically the motion you are interested in. Like $x(t)=a\;\sin(\omega t)$ or something similar.

 

[THE HARLEQUIN]

yes .. you can suppose that
the spacecraft moves along the straight line AB , starting from its midpoint while its path is defined by
x(t)=a\;\sin(\omega t)

 

[Dale]

Then you simply apply the formulas:
$v(t)=dx/dt=a \omega\; \cos(t)$
and
$\tau = \int \sqrt{1-v^2/c^2} dt=\frac{1}{\omega}\sqrt{1-k^2/c^2}E\left(t\omega|\frac{k^2}{k^2-c^2}\right)$
where $k=\omega a$ and $E$ is the complete elliptic integral function of the second kind.

 

[THE HARLEQUIN]

thank you for the reply ... it was a nice mathematical approach i assume but the problem is :

τ=∫1−v2/c2−−−−−−−−√dt=1ω1−k2/c2−−−−−−−−√E(tω|k2k2−c2)\tau = \int \sqrt{1-v^2/c^2} dt=\frac{1}{\omega}\sqrt{1-k^2/c^2}E\left(t\omega|\frac{k^2}{k^2-c^2}\right)
where k=ωak=\omega a and EE is the complete elliptic integral function of the second kind.

the equation you wrote above has gone twenty miles over my head ...
it will be nicer if you can make me visualize how the simple harmonic motion of the spacecraft affects the time using space-time diagram and
i have another question :
if another spaceship that is parallel to our original spaceship is moving not with a simple harmonic motion but with a constant speed along AB straight line ( just accelerating at the specific points A and B enough to make the direction of the velocity reverse) and reaching the border points at the same time as the original spaceship then what is the time measured by the rocket driving person for the second spaceship and how will it differ from the other two ?
( i mean if three persons of the same age stays in spacecraft 1 , spacecraft 2 and the rocket respectively , when the rocket comes back how will they differ in age from one another ? )

 

[Dale]

the equation you wrote above has gone twenty miles over my head

Perhaps you can ask a simpler question then. That is the answer to the question you asked, but the question you asked may have been needlessly complicated.

I am not going to draw a spacetime diagram of simple harmonic motion for you. You can see plots of that all over the internet, it is not worth the effort to plot it and post it here. When the spaceship is going fast, it is strongly time dilated, when it is going slow then it is less time dilated, when it is momentarily at rest then it is momentarily not time dilated.

i have another question

If your original question was too complicated for you to understand the answer then you should try a simpler question, not a more complicated question. You are just causing more work for me for no benefit to you. You are completely going about this the wrong way. If you don't understand something then you need to simplify the scenario to eliminate everything else besides one key thing that you don't understand.

Ask a simpler question, that you can understand the answers to.

 

[THE HARLEQUIN]

Perhaps you can ask a simpler question then. That is the answer to the question you asked, but the question you asked may have been needlessly complicated.

I am not going to draw a spacetime diagram of simple harmonic motion for you. You can see plots of that all over the internet, it is not worth the effort to plot it and post it here. When the spaceship is going fast, it is strongly time dilated, when it is going slow then it is less time dilated, when it is momentarily at rest then it is momentarily not time dilated.

If your original question was too complicated for you to understand the answer then you should try a simpler question, not a more complicated question. You are just causing more work for me for no benefit to you. You are completely going about this the wrong way. If you don't understand something then you need to simplify the scenario to eliminate everything else besides one key thing that you don't understand.

Ask a simpler question, that you can understand the answers to.

i never wanted any mathematical proof i just wanted concrete physical answers with graphical representation to visualize what really happens ...anyway thanks for your time though

 

[Dale]

You are welcome.

FYI, here is a diagram of simple harmonic motion:
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html

To turn it into a standard spacetime diagram simply flip it vertical instead of horizontal.

 

[THE HARLEQUIN]

May be i will have to ask the question again with some diagrams to clarify things because i am not getting proper answers .

 

[Dale]

Clarification and simplification would both help.