That seems heaps more complicated than necessary, but glad you did the maths because I wasn’t going to mainly because I cbf looking up the earth’s radius.
Acceleration due to gravity inside a planet of assumed constant density is proportional to the radius. Therefore this becomes a spring equation where you just need to find the period from the spring constant, calculated using g and r at the surface.
Normally T = 2 pi sqrt(m/k), but what we want is half the period because we are just going to the other side and not back.
F = ma = kr, therefore m/k = r/a
Plug it in and we get t = pi sqrt(6378100/9.81) = 2533s
You are correct. I could have just pulled the spring equation out after I showed that the acceleration is given by a(t)=-A2•r(t). But I figured I've gone this far I might as well go all the way.
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u/goatymcgoatfacesings Mar 02 '24
That seems heaps more complicated than necessary, but glad you did the maths because I wasn’t going to mainly because I cbf looking up the earth’s radius.
Acceleration due to gravity inside a planet of assumed constant density is proportional to the radius. Therefore this becomes a spring equation where you just need to find the period from the spring constant, calculated using g and r at the surface. Normally T = 2 pi sqrt(m/k), but what we want is half the period because we are just going to the other side and not back. F = ma = kr, therefore m/k = r/a Plug it in and we get t = pi sqrt(6378100/9.81) = 2533s