The Andromeda galaxy is at a distance of 2.1 X 10^22 m from our Galaxy. The mass of Andromeda is 6 X 10^41 kg and the mass of our Galaxy is 4 X 10^41 kg. (a) Gravity accelerates the galaxies toward each other. As reckoned in an inertial reference frame, what is the acceleration of Andromeda? What is the acceleration of our Galaxy? Treat both galaxies as point particles.
I know the answers are 6.0 X 10^-14 m/s^2, 9.1 X 10^-14 m/s^2; I just need help understanding how to solve these problems.
If we let the CG be the designated PE = 0 point and M the
equivalent mass at that point, then we'd have pe = mGM/(r - x) and
pe' = m'GM/x so that your TE = KE + pe + pe'; where x = the CG
position between the two at distance r apart. m and m' are the
point masses of the two galaxies.
I suggest this as the CG is the reference point for the speeds and consequent KE's. So it seems proper that it ought to also be the reference point for the PE. You can solve for x, the CG position between r, their distance and everything else is given. And you can solve for M by sum of torques = zero as the two galaxies are not spiraling around each other in this case.
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