In this problem, we show that the clock's length appears shortened when viewed from a reference frame
Question:
In this problem, we show that the clock's length appears shortened when viewed from a reference frame S in which the clock is moving with relative speed v to the right, parallel to its long axis. The key is to calculate explicitly the length of time that an observer in S will measure between "ticks" of the moving clock in terms of LS, the length of the clock in the S frame. Comparing this with the known period of the clock in the S frame allows us to compute the length of the clock in the S frame. Einstein used gedanken experiments such as this to clarify his thinking as he strove to remove inconsistencies from the basic formulations of physics.
Part B
Use time dilation to determine the time tS between the ticks of the light clock as viewed from reference frame S.
Express the time tS between these ticks in terms of L, the relative speed v, and the speed of light c
Answer:
We now come to the key step: calculating the round-trip travel time (the clock tick time) from the perspective of an observer in S. This is not trivial, because we know that the separation of the moving mirrors is L in the moving frame, but it is not necessarily L in the stationary reference frame. In fact, the point of this problem is to calculate the length LS of the clock in the system S. We do know that the pulse of light will travel at speedc relative to the coordinate system S. Therefore our job is to find the time tS it takes to catch the moving mirror (which appears to be LS ahead of the detector mirror) and return to the mirror at the end of the clock with the detector mirror.
Part E
Notice that the clocks at the front and rear mirrors of the light clock do not show the same time when viewed from the moving frame. Why is this?
a) The front clock experiences a greater time dilation because it moves ahead of the rear clock. |
b) The clocks start simultaneously in the rest frame, so they cannot start simultaneously in the moving frame. |
c) Since the light clock is passing the observer, the signal from the front clock takes longer to reach the observer, making the time appear earlier. |
Vector Mechanics for Engineers Statics and Dynamics
ISBN: 978-0073212227
8th Edition
Authors: Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell