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ID: 2125814 • Letter: B
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In the Hafele-Keating experiment (see notes), the time-dilation effect of special relativity was tested by sending two jets around the world, one traveling East in the direction of the Earth's rotation, and the other traveling West opposite to the direction of the Earth's rotation. Assume for simplicity that they travel around the equator without stopping at speed vP = 400 mph and return to their starting point. Ignoring the general relativistic effects due to the elevation of the planes above the ground (which are weaker than the special relativistic effects), estimate the differencedelta tE - delta tG in the time elapsed for the East bound clock compared to a clock which remained at the initial point on the ground the entire time. (Note that the Earth's radius is 3959 mi or 6,371 km.) Hint: first calculate the time elapsed delta t0 in a "rest-frame" O corresponding to the North pole by taking into account the speed vp of the Eastbound plane relative to the ground, the speed vo of the ground around the Earth's axis, and the circumference of the Earth. Then, following the notes, write expressions for the time elapsed delta tG on the ground and the time elapsed delta tE on the traveling (Eastbound) clock as a function of vo, vp, and A to- Subtract these two expressions to obtain an expression for AtE - Atc, as a function of vP, VQ, and delta to. Then use this expression to estimate the value of delta tE - delta tG. Note: the expressions in the notes for delta tG and delta tE are based on the approximations and which are valid for small u/c. In a simple thought experiment, Einstein showed that there is mass associated with electromagnetic radiation. Consider a box of length L and mass M resting on a frictionless surface. At the left wall of the box is a light source that emits radiation of energy E, which is absorbed at the right wall of the box. According to classical electromagnetic theory, this radiation carries momentum of magnitude p = E/c. (a) Find the recoil velocity of the box such that momentum is conserved when the light is emitted. (Since p is small and M is large, you may use classical mechanics.) (b) When the light is absorbed at the right wall of the box, the box stops, so the total momentum remains zero. If we neglect the very small velocity of the box, the time it takes for the radiation to travel across the box is delta t = L/c. Find the distance moved by the box in this time, (c) Show that if the center of mass of the system is to remain at the same place, the radiation must carry' mass m = E/c2.Explanation / Answer
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