The Relativistic Particle Accelerator Ring · AI Generator #GEN-DCCC10EC
Physics
Problem
A new particle accelerator uses a clever design where particles travel in a square ring rather than a circular one. The ring has four straight segments, each of length L=2.0km in the lab frame, connected by very short curved sections that can be neglected. Protons are accelerated to a speed v=0.80c and travel around this square ring.
Due to the straight segments, engineers can place detector stations at the midpoint of each segment to measure timing. In the lab frame, detectors are synchronized and measure the time intervals between when a proton passes consecutive detector stations.
(a) What time interval Δtlab does the lab measure between the proton passing two consecutive detector stations?
(b) In the proton's rest frame, what is the length L′ of each straight segment of the ring?
(c) According to the proton's wristwatch (proper time), what time interval Δτ elapses between passing two consecutive detector stations?
(d) The proton carries an internal "clock" (its proper time). After completing exactly 1000 loops around the entire ring, how much less time has elapsed on the proton's clock compared to the lab clocks?