1.11 Before doing, this problem, read footnote 4 in section 1.3 about linear hom
ID: 1878308 • Letter: 1
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1.11 Before doing, this problem, read footnote 4 in section 1.3 about linear homogeneous DEQs. A particle far from the Earth feels the pull of gravity GMm 2, where Gis the gravitational constant, M the mass of the Earth, Waves and Oscillations m the mass of the particle, and x is the distance from the center of the Earth to the particle. Assume that the particle moves along a straight line that passes through the center of the Earth, that gravity is the only force acting on it, and that for the time period we are considering the particle doesn't touch the surface of the Earth (i.e., that it's out in space). (a) Write a DEQ of motion for the particle. (As an example of what ioucan b' bero f ation, for the simpl harmonte oweliltori (b) If x1 (t) is one solution of the DEQ and x2 (1) is another, is the combination x1 +x2 necessarily a solution? Why or why not?Explanation / Answer
force of gravity is attractive in nature.
total force=-G*M*m/x^2
==>m*x''=-k/x^2
==>x''+(k/(m*x^2))=0
this is the DEQ for the motion.
part b:
if x1 is one solution, then it satisfies the DEQ
==>x1''+(k/(m*x1^2))=0...(1)
x2 is also another solution, then it also satisfies the DEQ
==>x2''+(k/(m*x2^2))=0...(2)
adding equations 1 and 2,
(x1+x2)"+(k/(m*x1^2))+k/(m*x2^2))=0...(3)
if x1+x2 s a solution,it has to satisfy the DEQ
that means, (x1+x2)"+(k/(m*(x1+x2)^2)=0...(4)
equations 3 and 4 are not equivalent.
hence x1+x2 is not a solution.
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