Measurement of length

Length is the distance between any two points in space; its SI unit is the metre. Objects of interest range from the macrocosm (galaxies, stars, planets -- a large world of large objects and large distances) to the microcosm (molecules, atoms, protons, electrons, bacteria -- a small world of small objects and small distances), so different techniques are needed at different scales.

Direct measurement (roughly $10^{-5}$ m to $10^2$ m): a metre scale measures $10^{-3}$ m to 1 m; a vernier caliper extends this down to about $10^{-4}$ m (least count 0.1 mm); a screw gauge goes further, to about $10^{-5}$ m (least count 0.01 mm), for objects up to roughly 50 mm.

• Screw gauge: magnifies a small linear motion into the easily-read circular motion of a screw. A reading is built from the pitch scale reading (PSR) -- the last visible mark on the linear scale -- plus the head scale coincidence (HSC) -- the head-scale division that lines up with the reference line -- multiplied by the least count. Example: PSR = 6 mm, HSC = 40 divisions, least count 0.01 mm $\Rightarrow$ reading $=6\ \text{mm}+(40\times0.01\ \text{mm})=6.40$ mm.
• Vernier caliper: measures dimensions like the diameter or depth of a hole, from the main scale reading (MSR) plus the vernier scale coincidence (VSC) -- the vernier division that lines up exactly with a main-scale division -- multiplied by the least count. Example: MSR = 2.2 cm, VSC = 4 divisions, least count 0.01 cm $\Rightarrow$ reading $=2.2\ \text{cm}+(4\times0.01\ \text{cm})=2.24$ cm.

Both instruments can carry a zero error: if the reading is non-zero when the jaws/anvil are fully closed, that offset (positive or negative) must be added to or subtracted from every subsequent reading (Figure 1.2 shows the no-error, positive-error and negative-error cases for both instruments).

Indirect measurement of large distances (heights, and astronomical distances neither a scale nor a caliper can reach):

• Triangulation method (for the height of an accessible object, Figure 1.3): stand at a point $C$, a known horizontal distance $x$ from the base of a tower/tree of height $h=AB$, and measure the angle of elevation $\theta=\angle ACB$ to the top with a range finder. From the right triangle, $\tan\theta=\dfrac{h}{x}$, so $h=x\tan\theta$.
• Parallax method (for the distance of a planet or star, Figure 1.4): parallax is the apparent shift in an object's position against a distant background when viewed from two different points; the separation between those two viewing points is the basis, $b$. If the basis subtends a small parallax angle $\theta$ at the object, and $x$ is the (large) distance to the object, then treating the basis as a small arc of a circle of radius $x$: $\theta=\dfrac bx$, so $x=\dfrac{b}{\theta}$ once $b$ and $\theta$ are known.
• Distance of the Moon from the Earth (Figure 1.5): observe the Moon simultaneously from two diametrically opposite points $A$, $B$ on Earth's surface (separation $AB=b$, the Earth's diameter), each against the same distant background star; the two individual parallax angles $\theta_1,\theta_2$ add up to the Moon's total parallax $\theta=\theta_1+\theta_2$. Since $AM\approx MC$ (the Earth-Moon distance), $\theta\approx\dfrac{AB}{MC}$, giving $MC=\dfrac{AB}{\theta}$. …