which of ωI and ωS is greater. The reason for the sign change is that when ωI > ωS, the spheres actually are moving faster than the rotating observers measure, so they measure a tension in the string that actually is larger than they expect; hence, the fictitious force must increase the tension (point outward). When ωI < ωS, things are reversed so the fictitious force has to decrease the tension, and therefore has the opposite sign (points inward). Is the fictitious force ad hoc? The introduction of FFict allows the rotational observers and the inertial observers to agree on the tension in the string. However, we might ask: "Does this solution fit in with general experience with other situations, or is it simply a "cooked up" ad hoc solution?" That question is answered by seeing how this value for FFict squares with the general result (derived in Fictitious force): The subscript B refers to quantities referred to the non-inertial coordinate system. Full notational details are in Fictitious force. For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres: and the velocity of this sphere as seen in the rotating frame: where uθ is a unit vector perpendicular to uR pointing in the direction of motion. The frame rotates at a rate ωR, so the vector of rotation is Ω = ωR uz (uz a unit vector in the z-direction), and Ω × uR = ωR (uz × uR) = ωR uθ ; Ω × uθ = −ωR uR. The centrifugal force is then: which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is and has the ability to change sign, being outward when the spheres move faster than the frame ( ωS > 0 ) and being inward when the spheres move slower than the frame ( ωS < 0 ). Combining the terms: Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ad hoc solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ωI, ωS is the greater, inasmuch as the centrifugal force contribution always is