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It is claimed in [1] that the Moon always falls towards the Sun? Can this be so? It suffices to suppose that the Earth’s orbit about the Sun, and the Moon’s orbit about the Earth, are both circular. 2 Solution The term “fall ” apparently means different things to different people. The usual convention is that an object is “falling ” (down) if its velocity is “downwards. ” However, some people consider that an object is “falling ” (down) if its acceleration is “downwards ” even when its velocity is “upwards.” In cylindrical coordinates (r, φ, z) the position vector r = r ˆr + z ˆz has velocity vector where ˙r = dr/dt, etc., and acceleration vector v = ˙r =˙r ˆr + r ˙ φ ˆ φ +˙z ˆz, (1) a = ¨r =(¨r − r ˙ φ 2) ˆr +(r ¨ φ +2˙r ˙ φ) ˆ φ +¨z ˆz, (2) Taking the Earth’s and Moon’s orbits to be circular relative to the Sun and Earth, with angular velocities Ω and ω, respectively, the distance r from the Sun to the Moon obeys r 2 = r 2 1 + r2 2 +2r1r2 cos[(ω − Ω)t], (3) where r1 is the radius of the Earth’s orbit, r2 <r1 is the radius of the Moon’s orbit, and the Moon is at its maximal distance from the Sun at time t = 0. The time derivative of eq. (3) is r ˙r = −(ω − Ω)r1r2 sin[(ω − Ω)t]. (4) Thus, the radial velocity ˙r ˆr oscillates in sign and by the usual convention the Moon does not always fall towards the Sun. 2.1 Radial Acceleration The acceleration a of the Moon (whose position vector is r) follows from Newton’s law of gravitation (ignoring effects of other planets and stars) as a = − GMSun r

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Year: 2010

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