What if one used polar coordinates in the two frames? In frame S the radial motion is constant and there is no angular motion. Hence, the acceleration is: a = ( r ¨ ¨ − − r θ θ ˙ ˙ 2 ) r ^ ^ + ( r θ θ ¨ ¨ + 2 r ˙ ˙ θ θ ˙ ˙ ) θ θ ^ ^ = 0 , and each term individually is zero because θ θ ˙ ˙ = 0 , θ θ ¨ ¨ = 0 and r ¨ ¨ = 0 . There is no force, including no r θ θ ˙ ˙ 2 "force" in frame S. In frame S' , however, we have: a ′ = ( r ¨ ¨ ′ − − r ′ θ θ ˙ ˙ ′ 2 ) r ^ ^ ′ + ( r ′ θ θ ¨ ¨ ′ + 2 r ˙ ˙ ′ θ θ ˙ ˙ ′ ) θ θ ^ ^ ′ . In this case the azimuthal term is zero, being the rate of change of angular momentum. To obtain zero acceleration in the radial direction, however, we require: r ¨ ¨ ′ = r ′ θ θ ˙ ˙ ′ 2 . The right-hand side is non-zero, inasmuch as neither r ′ nor θ θ ˙ ˙ ′ is zero. That is, we cannot obtain zero force (zero a ′ ) if we retain only r ¨ ¨ ′ as the acceleration; we need both terms.