02 / CHAPTER 10
Non-inertial frames
Use centrifugal and Coriolis terms carefully when the frame demands them.
ANCHOR EQUATION
Rotating-frame acceleration
In symbols: a = arel + 2ω×vrel + ω×(ω×r)
Non-inertial terms are bookkeeping for observing motion from an accelerating frame.
A rotating frame adds apparent terms
An observer in an accelerating or rotating frame can still use Newton’s law if extra terms account for the frame’s acceleration. These apparent forces are bookkeeping effects tied to the observer, not new interactions with a third body.
Transform the acceleration
If inertial position is r = R(t) + r′, then a = A + a′ for a translating frame, so m a′ = F − mA. The term −mA is the translational inertial force. For a frame rotating with angular velocity ω, differentiating rotating vectors adds ω × terms, producing centrifugal −mω × (ω × r′) and Coriolis −2mω × v′ terms in the rotating-frame balance.
At latitude where Earth’s rotation matters, a parcel moving north with speed 10 m/s experiences a Coriolis acceleration of order 2Ωv ≈ 2(7.29×10⁻⁵)(10) = 1.46×10⁻³ m/s², tiny for a room but important over long distances.
Choose the simplest frame
An inertial frame avoids apparent terms but may be awkward for a rotating platform. The apparent-force model is valid only if every relevant frame acceleration term is included and the sign convention is consistent.
Work it: A 2 kg mass sits 0.5 m from a platform axis rotating at 4 rad/s. Calculate the centrifugal force in the platform frame and identify its direction. What changes if the mass moves radially?