Question

4. Consider a satellite of mass m moving in a circular orbit around the Earth at a constant speed v and at an altitude h above the Earth's surface as illustrated in the figure. (a) Determine the speed of the satellite in terms of g, h, Re (the radius of the Earth), and Me (the mass of the Earth). (b) If the satellite is to be geosynchronous (that is, appearing to remain over a fixed position on the Earth), how fast is it moving through space?[Use google/ebook to find the mass of the Earth]

Answer 1

*Solution - .RE 10 fig-2) (a). Speed of the Satellite: -- From the Newton's law of gravitation, the gravitation-al force between the earth and the satellite is given by - F=G MEM 1 r where, G= universal gravititional constont ME= mass of the earth m = mass of the satellite 8= distance of satellite from the centre of the earth. when a satellite moving around the earth with velocity to then centripetal force is given as @ F = mez z

From equation ☺ onde, we get GMEM mze? 8 7 2 Gre From fig-(1) , If x= (Reth), then v= Gre (Reth) Ans. (b). period of revolution of satellite'- The time period of a satellite around the earth is given by T = 2 g v 08, T = 2 IT GME (Reth) or; T = 2017 | Reth) GME since, GME = GRE , then T= 21 (Reth) GRE whese, g= acceleration due to gravity

For geosynchronous the time period will be T= 23h, 56 min ; 4 sec : TE12360*60)+(56860) ++ T= 86164 seconds then T= 2IT CREth3 GRE or, T²= = 412 (Re+h)3 GRE o*i (work)= 13 2 т“g Re -2 47²

"13 oy, h RE та R- 4T12 (86016473 x(9.8)x(6.37 210093 4x (3114) or, 6:37X10 or h - 25+6)x167 (7424234896)*(9.8)x(40.5769)x10' (39.4384) 6:37 X10 or h 6 6:37 X10 2952273882125x1027 39.4384 821258189 or, ħ = [14857 851285:22107'2 6:37 Xio or ha 4214.497 X 104- 6.37x106 or, h = 42.145 X106 6:37 X106 or, 35.775 X106 m. h or th= 35775 km

then 8= Reth Y = 6137x10 + 35.775 X106 8 = 42:145 X90%. Now we com calculate the speed of satellite as - the geo synchronous v = 2118 T 20 - 6 2*3:14 X 42.145X10 86164 v= X106 264.6706 86164 0.003071707 x 106 ve = 3.07 x103 misec v Aug