i) Explain the use of screw gauge and vernier caliper in measuring smaller distances. ii) Write a note on tria

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Q. i) Explain the use of screw gauge and vernier caliper in measuring smaller distances.

ii) Write a note on triangulation method and radar method to measure larger distances.

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Concept understanding — Measurement of Small Distances (Screw Gauge and Vernier Caliper)

For lengths from about $10^{-5}$ m up to $10^2$ m, length can be measured directly: a metre scale for $10^{-3}$ m to 1 m; a vernier caliper for finer work, down to about $10^{-4}$ m (least count 0.1 mm); and a screw gauge for the finest direct work, down to about $10^{-5}$ m (least count 0.01 mm), for objects up to roughly 50 mm.

Screw gauge. Works by magnifying a small linear motion into the easily-read circular motion of a screw. A reading combines the pitch scale reading (PSR) -- the last fully-visible mark on the linear scale -- with the head scale coincidence (HSC) -- the head-scale division exactly aligned with the reference line -- times the least count:

\text{Reading}=\text{PSR}+(\text{HSC}\times\text{least count}).

Worked example: PSR $=6$ mm, HSC $=40$ divisions, least count $=0.01$ mm $\Rightarrow$ reading $=6+(40\times0.01)=6.40$ mm.

Vernier caliper. Measures the diameter or depth of a hole via the main scale reading (MSR) plus the vernier scale coincidence (VSC) -- the vernier division that lines up exactly with a main-scale division -- times the least count:

\text{Reading}=\text{MSR}+(\text{VSC}\times\text{least count}).

Worked example: MSR $=2.2$ cm, VSC $=4$ divisions, least count $=0.01$ cm $\Rightarrow$ reading $=2.2+(4\times0.01)=2.24$ cm.

Zero error. If a reading is non-zero when the jaws/anvil are fully closed, this offset (positive or negative) must be added to, or subtracted from, every subsequent reading before it is trustworthy.

Comparing the two. The screw gauge's least count (0.01 mm) is finer than the vernier caliper's (0.1 mm) -- i.e. a smaller least count means a more precise instrument, because it can resolve a smaller change. A screw gauge with pitch 1 mm and 100 head-scale divisions has least count $=\dfrac{1\text{ mm}}{100}=0.01$ mm, ten times finer than a vernier caliper whose sliding scale has 20 divisions over, say, 1 mm main-scale spacing (least count typically $0.1/20$ or similar, always coarser than the screw gauge in this comparison) -- confirming the screw gauge is the more precise of the two for small-object measurements.

Step 1 (Screw gauge). A screw gauge magnifies a small linear motion into the circular motion of a screw, and can measure objects up to about 50 mm with a least count of 0.01 mm. A reading combines the pitch scale reading (PSR) -- the last visible mark on the linear scale -- with the head scale coincidence (HSC) -- the head-scale division aligned with the reference line -- times the least count: $\text{Reading}=\text{PSR}+(\text{HSC}\times\text{LC})$ . Example: PSR $=6$ mm, HSC $=40$ , LC $=0.01$ mm $\Rightarrow6+(40\times0.01)=6.40$ mm.

Step 2 (Vernier caliper). A vernier caliper measures a hole's diameter or depth via the main scale reading (MSR) plus the vernier scale coincidence (VSC) -- the vernier division aligned with a main-scale division -- times the least count (typically 0.01 cm): $\text{Reading}=\text{MSR}+(\text{VSC}\times\text{LC})$ . Example: MSR $=2.2$ cm, VSC $=4$ , LC $=0.01$ cm $\Rightarrow2.2+(4\times0.01)=2.24$ cm. Both instruments may carry a zero error (non-zero reading when fully closed), which must be added/subtracted from every subsequent reading.

Step 3 (Triangulation method). For the height $h=AB$ of an accessible tower/tree: stand at $C$ , a known horizontal distance $x$ from the base $B$ , measure the angle of elevation $\theta$ to the top $A$ with a range finder; from the right triangle, $\tan\theta=h/x$ , so $h=x\tan\theta$ .

Step 4 (Parallax and RADAR for larger distances). For very large (astronomical) distances, parallax uses two widely-separated observation points and the relation $x=b/\theta$ ( $b$ = basis, $\theta$ = parallax angle). RADAR beams a radio pulse to a nearby planet and times the round-trip $t$ of its echo: since the wave covers the distance twice, $d=\dfrac{v t}{2}$ , with $v$ the speed of the radio wave. This method also gives an aircraft's altitude.

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Q.i) Explain the use of screw gauge and vernier caliper in measuring smaller distances.

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