Next, the same approach is used to find the fictitious forces of a (non-inertial) rotating frame. For example, if a rotating polar coordinate system is adopted for use in a rotating frame of observation, both rotating at the same constant counterclockwise rate Ω, one can find the equations of motion in this frame as follows: the radial coordinate in the rotating frame is taken as r, but the angle θ' in the rotating frame changes with time: θ θ ′ = θ θ − − Ω Ω t . Consequently, θ θ ˙ ˙ ′ = θ θ ˙ ˙ − − Ω Ω . Plugging this result into the acceleration using the unit vectors of the previous section: d 2 r d t 2 = [ r ¨ ¨ − − r ( θ θ ˙ ˙ ′ + Ω Ω ) 2 ] r ^ ^ + [ r θ θ ¨ ¨ ′ + 2 r ˙ ˙ ( θ θ ˙ ˙ ′ + Ω Ω ) ] θ θ ^ ^ = ( r ¨ ¨ − − r θ θ ˙ ˙ ′ 2 ) r ^ ^ + ( r θ θ ¨ ¨ ′ + 2 r ˙ ˙ θ θ ˙ ˙ ′ ) θ θ ^ ^ − − ( 2 r Ω Ω θ θ ˙ ˙ ′ + r Ω Ω 2 ) r ^ ^ + ( 2 r ˙ ˙ Ω Ω ) θ θ ^ ^ . The leading two terms are the same form as those in the inertial frame, and they are the only terms if the frame is not rotating, that is, if Ω=0. However, in this rotating frame we have the extra terms: