397_Physics ProblemsTechnical Physics

397_Physics ProblemsTechnical Physics - 399 Chapter 13...

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Chapter 13 399 Additional Problems P13.47 Let m represent the mass of the spacecraft, r E the radius of the Earth’s orbit, and x the distance from Earth to the spacecraft. The Sun exerts on the spacecraft a radial inward force of F GM m rx s s E = bg 2 while the Earth exerts on it a radial outward force of F GM m x E E = 2 The net force on the spacecraft must produce the correct centripetal acceleration for it to have an orbital period of 1.000 year. Thus, FF GM m GM m x mv m T SE S E E EE E −= = L N M M O Q P P 22 2 2 2 π which reduces to GM GM x T S E E E 2 2 4 . (1) Cleared of fractions, this equation would contain powers of x ranging from the fifth to the zeroth. We do not solve it algebraically. We may test the assertion that x is between 1 47 10 9 . × m and 148 10 9 . × m by substituting both of these as trial solutions, along with the following data: M S 1991 10 30 . k g , M E 5983 10 24 k g , r E 1496 10 11 . m, and T == × 1000 3156 10 7 ..
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