question_answer 61)

                  A satellite is to be placed in equatorial geostationary orbit around earth for communication.                 (a) Calculate height of such a satellite.                 (b) Find out the minimum number of satellites that are needed to cover entire earth so that at least one satellites is visible from any point on the equator.             [\[M=6\times {{10}^{24}}\,\,kg,\,R=6400\,km,\,\]\[25th,\,\,G=6.67\]\[\times {{10}^{-11}}\,SI\]units]

Answer:

                  (a) Refer Article 8.11 of chapter 8 of New Millennium Physics, Class XI.                 (b) From the figure,                 \[\cos \,\theta =\,\frac{R}{R+h}\,\,=\,\frac{1}{\left( 1+\frac{h}{R} \right)}\]                 Now\[\frac{h}{R}\,=\,\frac{35930}{6400}\,\,=\,5.61\]                 \[\therefore \] \[\cos \,\,\theta =\,\frac{1}{6.61}\,\,=\,0.1513\] or\[\theta =\,{{81.29}^{o}}\]                 Let \[n=\] number of satellites that are needed to cover entire earth.                 \[\therefore \] \[n=\,\frac{{{360}^{o}}}{2\theta }\,=\,\frac{{{360}^{o}}}{{{162.59}^{o}}}\,\,=\,2.21\,\simeq \,3\]