Accelerating coordinates require inertial-force terms

Let an inertial observer describe position r, and let the train's origin have position R. If the train's axes stay parallel to the inertial axes, the passenger's relative position is r minus R. Differentiate twice: the passenger measures the physical object's inertial acceleration minus the train origin's acceleration. Substituting this relation into Newton's law leaves the usual interaction-force sum and an additional term equal to minus mass times origin acceleration. This term depends on the coordinate system and acts on every mass in proportion to its inertial mass.

The inertial term does not have a third-law partner on a second object because it is not a new interaction. A real seat or floor force can nevertheless be needed to hold a passenger stationary relative to the accelerating vehicle. In the inertial description that contact force produces the passenger's acceleration. In the vehicle description it balances the inertial term. These are two coordinate descriptions of one contact force and one trajectory. Mixing the vehicle's zero relative acceleration with the inertial-frame force equation would incorrectly conclude that the contact force must vanish.

A rotating basis changes even a vector whose physical direction is fixed. Its inertial derivative equals its rotating-frame derivative plus the angular velocity crossed into that vector. Applying this relation twice gives terms proportional to relative velocity, position, and the time derivative of angular velocity. Their negatives appear on the force side of the rotating-frame equation. Keep the frame's angular velocity and origin acceleration distinct: one describes orientation change, while the other describes translation. A merry-go-round can rotate about an origin whose translational acceleration is zero.

The centrifugal term points outward from a fixed rotation axis. The Coriolis term depends on velocity measured in the rotating frame and vanishes for an object fixed in that frame. The Euler term appears when the frame's angular velocity changes. For constant rotation it vanishes, but centrifugal and Coriolis terms may remain. These terms are not interchangeable labels for any curved motion. In the inertial description of a body in circular motion, an inward resultant interaction force supplies centripetal acceleration; no centrifugal interaction must be added to cancel it in that inertial equation.

An effective gravity can combine actual gravitational acceleration with a position-dependent centrifugal term when analyzing objects nearly stationary on a steadily rotating platform. It does not absorb the velocity-dependent Coriolis term for moving objects. Similarly, apparent weight measured by a scale is a contact force, not a direct reading of gravitational attraction alone. An upward-accelerating lift increases the scale reading; a freely falling lift gives zero support force in the ideal point-particle model even though gravitational acceleration remains nonzero in the nearby inertial approximation.

A reliable frame audit names the observer, defines which velocities are relative, and checks a case with no interactions. A free particle moves along a straight line at constant speed in an inertial frame but can trace a curved path in rotating coordinates. Both descriptions must transform into each other. In laboratory work, Earth's rotation can be neglected when its accumulated effects are below the desired uncertainty; it becomes important for long flight times, sensitive pendulums, and large-scale flows. Frame choice is a convenience, while the accuracy requirement decides which transformation terms can safely be omitted.