= Specific force
{title2=$\mathbf f=\mathbf a-\mathbf g$}
\b[Specific force] is the nongravitational force per unit mass experienced by a body. In a Newtonian local model, neglecting Earth's rotation, let $\mathbf a$ be its coordinate <acceleration> and $\mathbf g$ the local gravitational acceleration vector. <Newton's second law> gives
$$
m\mathbf a=m\mathbf g+\mathbf F_{\mathrm{ng}},\qquad
\mathbf f=\frac{\mathbf F_{\mathrm{ng}}}m=\mathbf a-\mathbf g.
$$
Specific force has units $\mathrm{m\,s^{-2}}$. An ideal <accelerometer> senses this quantity along its own axes, rather than coordinate acceleration alone. The Android sensor documentation illustrates the distinction with a nonzero stationary-table reading and approximately zero reading during <free fall>.<Motion sensors in the Android Developers documentation>[Use the accelerometer, table and free-fall examples]{ref}
Choose an upward unit vector $\widehat{\mathbf z}$ and $\mathbf g=-g\widehat{\mathbf z}$. A stationary supported instrument has $\mathbf a=0$ and therefore $\mathbf f=+g\widehat{\mathbf z}$: support supplies the upward nongravitational force. In ideal <free fall>, $\mathbf a=\mathbf g$ and $\mathbf f=0$. Gravity has not disappeared; the instrument and its <proof mass> accelerate together.
For an original elevator example with $g=9.81\,\mathrm{m\,s^{-2}}$, an upward coordinate acceleration of $2.00\,\mathrm{m\,s^{-2}}$ gives upward specific force $11.81\,\mathrm{m\,s^{-2}}$. Moving upward at constant velocity instead gives $9.81\,\mathrm{m\,s^{-2}}$. The reading depends on acceleration and support, not the direction of velocity.
If a <rotation matrix> $R$ maps instrument-axis components to world-axis components, an ideal body-axis readout is $R^\mathsf T(\mathbf a-\mathbf g)$. Reconstructing $\mathbf a$ therefore also requires orientation and a gravity model. A sustained horizontal acceleration can resemble a tilted gravity-related reading; an accelerometer alone does not generally separate those causes. In this local slow-motion setting the sensed support acceleration connects to <proper acceleration>.
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