Question

Many communication satellites are placed in a circular orbit around the Earth at a radius where the period (the time to go around the Earth once) is 24 hours. If the satellite is above some point on the equator, it stays above that point as the Earth rotates, so that as viewed from the rotating Earth the satellite appears to be motionless. That is why you see dish antennas pointing at a "fixed" point in space.

(a) Calculate the radius of the orbit of such a "synchronous" satellite. Show your work on paper, starting from the Momentum Principle (don't just quote some canned formula).

(b) Electromagnetic radiation including light and radio waves travels at a speed of 3 × 10⁸ m/s. If a phone call is routed through a synchronous satellite to a friend in a nearby city, what is the minimum delay between saying something and getting a response? Note that you're standing on the surface of the Earth, not at its center. Show your work on paper, and include a diagram of the situation.

(c) Some human-made satellites are placed in "near-Earth" orbit, just high enough to be above almost all of the atmosphere. Calculate how long it takes for such a satellite to go around the Earth once. Because the atmosphere is very thin, you can make the approximation that the radius of the orbit is approximately the radius of the Earth. Show your work on paper.

(d) Calculate the orbital speed for a near-Earth orbit, which must be provided by the launch rocket. Show your work on paper. (Recently large numbers of near-Earth communications satellites have been launched. Their advantages include making the signal delay unnoticeable, but with the disadvantage of having to track the satellites actively and having to use many satellites to ensure that at least one is always visible over a particular region.)

(e) When the first two astronauts landed on the Moon, a third astronaut remained in an orbiter in circular orbit near the Moon's surface. During half of every complete orbit, the orbiter was behind the Moon and out of radio contact with the Earth. On each orbit, how long was the time when radio contact was lost? Show your work on paper.

Answer 1

A geosynchronous orbit  is an orbit around Earth of a satellite with an orbital period that matches Earth's rotation about its axis.

(a) For circular orbit , $F_c=F_g$ , i.e, centripetal force is balanced by gravitational attraction

M,v2 GMM 2 V=1 GM Vr

so, time period of rotation,

T = 291 = Tº = 47

  

= T° -

GMT2 472

on putting the values , we get r=42,160 Km that is

T = 6.6R

(b) for time delay we can assume the distance of satellite from earth surface will be
d 5.6RE
,therefor total distance traveled by the signal will be
4d 5.6 x 4R
,

So, time taken by signal to come back will be ,

4d 22.4Re – 0.478 sec c 3 108 = 0.478 sec

(c) For calculation of time period of rotation we use the formula of T

  

= T° -

on solving the equation taking r= radius of earth

T1.4hrs