Foundations — Inertial navigation — accelerometer measures non-gravitational specific force
Before you can read the parent note, you need to own every symbol it throws at you. We build them one at a time, from nothing, each one earning its place before the next appears.
1. A quantity with a direction: the vector (bold letters)
The plain number (a scalar) can only say "five." But "five, pointing up" needs an arrow. In navigation everything has a direction — forces pull one way, motion happens another way — so we cannot use plain numbers.
Components. To do arithmetic we write a vector as a list of numbers, one per direction (axis): Here is "how much of the arrow points along the x-axis," and so on. The little subscript letter just names which axis. This is why the parent note writes things like — it means "zero sideways, zero forward, and pointing down the z-axis."
2. Which way is "down"? The coordinate axes
Look at the figure: the green z-arrow points up. Gravity (the coral arrow) points the opposite way, so its component is , giving . The sign is not decoration — it is the entire reason a still accelerometer reads upward later. Keep the picture in mind whenever a sign appears.
3. Mass and force
We separate forces into two families, and this split is the heart of the topic:
| Family | Symbol | Picture | Touches the spring? |
|---|---|---|---|
| Contact force | a spring, a hand, a seat pressing on you | Yes | |
| Body force (gravity) | Earth tugging every atom at once | No |
4. Acceleration vs specific force
Why "per unit mass" (dividing by )? Because a heavier proof mass needs a bigger spring push for the same motion. Dividing by cancels the mass out, so the reading depends only on the situation, not on how big we built the sensor. That is why carries units — force () divided by mass ().
5. Gravity
Because we divided out mass, has the same units as and — which is what lets us subtract them cleanly in .
6. The inertial frame — where Newton's law is allowed
Why the topic insists on this: the parent note writes Newton's law "in an inertial frame" so that really is the true acceleration, not a fake one caused by the observer themselves accelerating. Get the frame wrong and extra phantom forces appear.
7. Newton's second law — the one equation everything comes from
Divide every term by and rearrange — this is the derivation the parent note performs:
\qquad\Longleftrightarrow\qquad \mathbf{f} = \mathbf{a} - \mathbf{g}$$ See [[Newton's Second Law]] for the full statement; here we only need this one form. ![[deepdives/dd-physics-3.5.13-d1-s03.png]] The figure shows the proof mass with just two arrows on it: the spring's contact push (lavender) and gravity's pull (coral). Newton says these two, added, equal $m\mathbf{a}$. Solve for the spring's push and you have exactly what the sensor reads. --- ## 8. The **integral** $\int dt$ — turning acceleration into position > [!definition] Integration $\int \,dt$ > The symbol $\int$ (a stretched "S," for "sum") with $dt$ means "**add up over tiny slices of time**." Integrating acceleration gives velocity; integrating velocity gives position. Picture stacking up thousands of tiny "$\text{speed} \times \text{tiny time}$" rectangles to get total distance travelled. $$\mathbf{a} \;\xrightarrow{\int dt}\; \mathbf{v} \;\xrightarrow{\int dt}\; \mathbf{r}$$ **Why the topic needs it:** the sensor gives acceleration-like data, but a navigator wants **position** $\mathbf{r}$. Integration twice is the bridge. This is why an early error in $\mathbf{a}$ balloons into a position error growing like $\tfrac12 g t^2$ — see [[Dead Reckoning and Error Drift]]. --- ## 9. Putting the symbols in order ```mermaid graph TD V["Vector = arrow with size and direction"] --> C["Components a_x a_y a_z"] C --> AX["Axes with z pointing up"] AX --> G["Gravity g points down so minus z"] M["Mass m"] --> F["Force F push or pull"] F --> CF["Contact force spring touches"] F --> BF["Body force gravity no touch"] CF --> SF["Specific force f = Fc over m"] BF --> G V --> ACC["Acceleration a rate of velocity change"] IF["Inertial frame fair observer"] --> N2["Newtons second law m a = sum F"] ACC --> N2 SF --> N2 G --> N2 N2 --> EQ["f = a minus g"] EQ --> INT["Integrate twice to get position"] INT --> TOP["Inertial navigation"] ``` Read the map top-down: arrows and axes give us gravity's sign; the force split gives us specific force; Newton's law in an inertial frame welds them into $\mathbf{f}=\mathbf{a}-\mathbf{g}$; integration turns that into position. That final chain **is** the parent topic. --- > [!recall]- Quick self-check on the signs > Why does $\mathbf{g}=(0,0,-g)$ have a minus sign? > ::: Because we chose z pointing **up**, and gravity points **down** — the opposite direction — so its z-component is negative. > [!mnemonic] > **Bold = arrow, plain = number.** If a letter is bold ($\mathbf{f},\mathbf{a},\mathbf{g}$) it has a direction; if it's plain ($m$, $g$, $t$) it's just a size. --- ## Equipment checklist Read each question, answer out loud, then reveal. If any stump you, re-read that section before the parent note. What does a **bold** letter like $\mathbf{a}$ mean, and what does it look like? ::: A vector — a quantity with size AND direction, drawn as an arrow. What do the subscripts in $(a_x,a_y,a_z)$ tell you? ::: How much of the vector points along each axis (x, y, z). With z pointing up, how do you write gravity as components? ::: $\mathbf{g}=(0,0,-g)$ — the minus because down is the $-z$ direction. What is the difference between mass $m$ and force $\mathbf{F}$? ::: Mass is how much stuff (a plain number, kg); force is a push/pull (a vector, newtons). Which forces can a spring feel — contact or body forces? ::: Only contact forces; body forces like gravity pull every atom equally and never touch the spring. Define specific force $\mathbf{f}$ in one line. ::: The contact force per unit mass, $\mathbf{f}=\mathbf{F}_{\text{c}}/m$ — what the accelerometer actually reads. Why do $\mathbf{f}$, $\mathbf{a}$, and $\mathbf{g}$ all share units $\text{m/s}^2$? ::: Because each is a force (or acceleration) **per unit mass** — dividing by $m$ gives $\text{m/s}^2$, letting us subtract them. What is an inertial frame and why do we need it? ::: A non-accelerating, non-spinning observer; Newton's law $m\mathbf{a}=\sum\mathbf{F}$ is only exactly true there. State Newton's second law with the two forces here. ::: $m\mathbf{a}=\mathbf{F}_{\text{c}}+m\mathbf{g}$. What does integrating acceleration twice give you? ::: First velocity $\mathbf{v}$, then position $\mathbf{r}$. --- ## Connections - [[Newton's Second Law]] — the single equation this whole page prepares you to use. - [[Equivalence Principle]] — why gravity being a "body force" makes it invisible to the sensor. - [[Strapdown Inertial Navigation System]] — where the integration chain lives. - [[Dead Reckoning and Error Drift]] — what happens when the integrated $\mathbf{a}$ is slightly wrong. - [[3.5.13 Inertial navigation — accelerometer measures non-gravitational specific force (Hinglish)|Parent topic →]]