Accelerometer

An accelerometer measures proper acceleration, the acceleration associated with nongravitational support and applied forces on its sensing mass. A common design suspends a proof mass on springs inside a housing. Housing motion displaces the mass relative to the support; a capacitive accelerometer reads the resulting change in capacitance. At frequencies well below mechanical resonance, Hooke's law and Newton's second law give displacement magnitude $|x|\simeq m|a|/k$, with spring constant $k$.^[ref]

A simple calibration rotates a stationary sensor so one axis points up, sideways and down. Near Earth's surface, the axis reading changes approximately from $+g$ through zero to $-g$, where $g\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$. A supported stationary device therefore has a nonzero reading even though its position is constant; in ideal free fall, the mass and housing fall together and the reading approaches zero. This experimentally separates the measured support acceleration from coordinate acceleration.^[ref]

For a horizontal ideal sensor with $m=10^{-6}\mathrm{kg}$ and $k=1\mathrm{N}{\mathrm{m}}^{-1}$, an acceleration of $2\mathrm{m}{\mathrm{s}}^{-2}$ gives a displacement magnitude of $2\mu \mathrm{m}$. The readout converts a tiny mechanical displacement into an electrical signal; its scale, bias and frequency response must be calibrated.

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 givesSpecific force has units $\mathrm{m}{\mathrm{s}}^{-2}$. 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.^[ref]

Choose an upward unit vector $\hat{\mathbf{z}}$ and $\mathbf{g}=-g\hat{\mathbf{z}}$. A stationary supported instrument has $\mathbf{a}=0$ and therefore $\mathbf{f}=+g\hat{\mathbf{z}}$: 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 $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, an upward coordinate acceleration of $2.00\mathrm{m}{\mathrm{s}}^{-2}$ gives upward specific force $11.81\mathrm{m}{\mathrm{s}}^{-2}$. Moving upward at constant velocity instead gives $9.81\mathrm{m}{\mathrm{s}}^{-2}$. 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.