Accelerated Observers: Pion Motion in Lab Frame

[wam_mi]

Homework Statement

In a laboratory, a pion moves in a circular orbit of radius r and period T as measured
in the lab frame. How much proper time elapses during one revolution as seen by the
pion?

Homework Equations

Let period = T
Let radius = r
Let v = velocity

(1) v = (2 * pi * r) / T

Equation of time dilation
(2) T(proper time) = T / gamma
where gamma is the usual Lorentz factor

The Attempt at a Solution

This is an accelerated problem since the velocity of the pion particle is constantly changing (well, it's direction). I tried to plug in equation (1) into equation (2) but I realized gamma is not constant since the velocity is changing... so how should I approach this problem. Help!

Thank you!

 

[gabbagabbahey]

First, your equation for proper time doesn't look right to me...

Second, what is the fundamental difference between velocity and speed? Which one of these quantities is $\gamma$ actually dependent on?

Third, is this really all the information you are given? Are you not told how (or if) the speed of the particle varies along its orbit?

 

[wam_mi]

First, your equation for proper time doesn't look right to me...

Second, what is the fundamental difference between velocity and speed? Which one of these quantities is $\gamma$ actually dependent on?

Third, is this really all the information you are given? Are you not told how (or if) the speed of the particle varies along its orbit?

Hi there,

(i) Is it ture that the time dilation equation: Lab frame's time = Lorentz factor * Proper time?

(ii) hm, velocity is a vector, whereas speed is the magnitude of speed. If the particle is moving round a circle, is it right to say that its magnitude of velocity is constant since only the direction is altering? Can one then infer the Lorentz vector is also a constant since it depends only on the square of velocity?

(iii) That's all the information I get, it just says the particle is going in a circular orbit of radius r and period T as measured in the lab frame. It's asking me to compute the proper time.

 

[gabbagabbahey]

Hi there,

(i) Is it ture that the time dilation equation: Lab frame's time = Lorentz factor * Proper time?

No. Look up the definition of proper time.

(ii) hm, velocity is a vector, whereas speed is the magnitude of speed. If the particle is moving round a circle, is it right to say that its magnitude of velocity is constant since only the direction is altering? Can one then infer the Lorentz vector is also a constant since it depends only on the square of velocity?

If the particle is undergoing uniform circular motion, then yes, its speed will be constant and so will the Lorentz factor.

(iii) That's all the information I get, it just says the particle is going in a circular orbit of radius r and period T as measured in the lab frame. It's asking me to compute the proper time.

I would assume that the particle is undergoing uniform circular motion then (although you should double check that this is what your professor intended before handing in your assignment).