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Physics LibreTexts

12.2: Translational acceleration of a reference frame

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10.2.1.PNG
Figure 12.2.1: Inertial reference frame (unprimed), and translational accelerating frame (primed).

Consider an inertial system (xfix,yfix,zfix) which is fixed in space, and a non-inertial system (xmov,ymov,zmov) that is moving in a direction relative to the fixed frame such as to maintain constant orientations of the axes relative to the fixed frame, as illustrated in Figure 12.2.1. The fixed frame is designated to be the unprimed frame and, to avoid confusion the subscript fix is attached to the fixed coordinates taken with respect to the fixed coordinate frame. Similarly, the translating reference frame, which is undergoing translational acceleration, has the subscript mov attached to the coordinates taken with respect to the translating frame of reference. Newton’s Laws of motion are obeyed only in the inertial (unprimed) reference frame. The respective position vectors are related by

rfix=Rfix+rmov

where rfix is the vector relative to the fixed frame, rmov is the vector relative to the translationally accelerating frame and Rfix is the vector from the origin of the fixed frame to the origin of the accelerating frame. Differentiating Equation ??? gives the velocity vector relation

vfix=Vfix+vmov

where vfix=drfixdt, vmov=drmovdt and Vfix=dRfixdt. Similarly the acceleration vector relation is

afix=Afix+amov

where afix=d2rfixdt2, amov=d2rmovdt2 and Afix=d2Rfixdt2.

In the fixed frame, Newton’s laws give that

Ffix=mafix

The force in the fixed frame can be separated into two terms, the acceleration of the accelerating frame of reference Afix plus the acceleration with respect to the accelerating frame amov.

Ffix=mAfix+mamov

Relative to the accelerating reference frame the acceleration is given by

mamov=FfixmAfix

The accelerating frame of reference can exploit Newton’s Laws of motion using an effective translational force FtranFfixmAfix. The additional mAfix term is called an inertial force; it can be altered by choosing a different non-inertial frame of reference, that is, it is dependent on the frame of reference in which the observer is situated.


This page titled 12.2: Translational acceleration of a reference frame is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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