- 1 This Problem Demonstrates A Principle Similar To What Global Positioning System Gps Satellites Use Geostationary 1 (213.61 KiB) Viewed 15 times
(1) This problem demonstrates a principle similar to what Global Positioning System (GPS) satellites use. Geostationary satellites are always directly over the same place on Earth's surface. To do this, geostationary satellites need to be over Earth's equator so they don't drift north or south. They also need to be in circular orbits, so they don't drift east or west. Geostationary satellites also need to orbit Earth with an orbital period of 24 hours, so as Earth rotates, the satellite will stay over the same point on Earth's equator. Because of the force of Earth's gravity, any geostationary satellite must be 42,164 km from Earth’s center. Assume that Earth is a sphere of radius 6378 km. Suppose one geostationary satellite is above Earth's equator at a longitude of 0.00°. Suppose a second geostationary satellite is above Earth's equator at a longitude of 120.00° east. Both satellites simultaneously transmit identical radio signals that have the same wavelength and amplitude. Suppose an aircraft is flying along the equator, from 0.00° longitude toward the east. Cruising altitude for a jet airliner is 10,000 m (about 32,800 feet). This is much smaller than the radius of Earth (and even smaller than the height of a geostationary satellite), so for this problem it can safely be assumed this aircraft is moving on or just above) Earth's surface. Do not neglect the of the curvature of Earth, since it is significant here. It may help to remember the law of cosines, which is: l1² = 12 +132 – 2 12 13 cos(LI), where lı, 12, and 13 are the lengths of the sides of a triangle, and angle L, is the angle of the vertex opposite side 11. (a) A radio in the aircraft receives the signals from the satellites, which combine into one signal. When the aircraft is at a longitude of 40.00° east, it is at the position of the maximum with m= 100. What is the frequency of the combined signal in the aircraft's radio? (b) If the aircraft travels farther east than it was in part (1)(a), at what east longitude will the aircraft encounter the next maximum of the combined signal in the aircraft's radio? (c) How many kilometers east is the result of (1)(b) from the starting point described in part (1)(a)? (a) Suppose one of the satellites uses a circular dish antenna to transmit a microwave beam for communication at normal incidence to Earth. The beam has a wavelength of 1 mm, and the dish has a radius of 1 m. Assuming Earth's atmosphere has no effect on the beam, what is the diameter of the beam at Earth's surface? (e) Unpolarized light is passed through three successive polarizing filters, each with its transmission axis at 45° to the preceding filter. What percentage of light gets through? (Hint: The answer is not zero, paradoxical as it may sound. This is an illustration of the physics that is really happening here: the polarizing filters rotate the plane of polarization.] (The next page is blank, so you may write answers there. You may also write answers on this page.)