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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.

A capacitive accelerometer detects acceleration through a change in capacitance caused by motion of a suspended proof mass. In the ADXL335, a micromachined polysilicon structure and differential capacitors form the mechanical sensor and its electrical readout.^[ref]

For an ideal pair of parallel plates with gaps $g-x$ and $g+x$, $C_{+}-C_{-}\approx 2\epsilon Ax/g^2$ when $|x|\ll g$. Combining this with $x\approx -ma/k$ gives an approximately linear acceleration signal. The approximation requires small displacement and operation sufficiently below mechanical resonance. A differential measurement identifies the direction of displacement and rejects some common changes affecting both capacitors similarly.

A microelectromechanical system combines small mechanical structures with electrical sensing or actuation. Fabrication methods related to semiconductor device fabrication can create suspended beams, membranes and movable masses together with their electrodes. Sensors and actuators can consequently be integrated into compact devices.

A capacitive accelerometer illustrates the mechanism. A suspended proof mass moves relative to its support under acceleration. In a simple spring model, a slowly varying support acceleration $a$ produces a relative displacement $x\approx -ma/k$, where $m$ is the mass and $k$ is the spring constant. Measuring a change of capacitance then measures that displacement. The ADXL335 uses suspended polysilicon structures and a differential capacitor to turn motion into an electrical signal.^[ref]

For an idealized pair of parallel-plate capacitors with equal area $A$ and gaps $g\mp x$, the capacitance difference isThis model explains why a small mechanical displacement can produce a signed electrical measurement; it is not a detailed model of every commercial MEMS geometry.

A proof mass is an inertial element whose relative motion or required support force provides a measure of acceleration. In a spring-supported accelerometer, the instrument frame moves with the object being measured, while the proof mass responds through its inertia and suspension.

An ideal one-axis model is $m\ddot{x}+c\dot{x}+kx=-ma$, where $x$ is displacement relative to the frame, $m$ is the proof mass, $c$ is damping and $k$ is the spring constant. At frequencies well below resonance, $x\approx -ma/k$. A capacitive accelerometer measures this displacement electrically. The suspension determines sensitivity and bandwidth together; a more compliant spring gives greater static displacement but a lower natural frequency.^[ref]

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.