Physics 1st Paper · Classes 11-12 · Chapter 1 · 1.14

Screw Gauge, Spherometer and Balance.

Measure a thin wire with a metre scale and its edge falls between two marks, so you end up guessing. A screw gauge does the same job down to a thousandth of a centimetre. In this note you will see how a reading comes out of three instruments: the screw gauge, the spherometer and the balance.

1. The screw gauge, pitch and least count

The screw gauge measures very small lengths, such as the diameter of a wire or the thickness of a thin sheet. It has a fixed rod A and a turning screw S. The barrel H carries a linear millimetre scale, and the rim of the cylinder D carries a circular scale of 100 (or 50) divisions.

The distance the screw advances along the linear scale in one full turn is the screw pitch. For this gauge P = 1 mm = 0.1 cm.

With 100 divisions on the circular scale, one division of turn moves the screw one hundredth of a pitch. So the least count L.C. = screw pitch / number of circular-scale divisions = 0.1 cm / 100 = 0.001 cm.

Why not pack in far more divisions for an even finer reading? Because divisions that close together cannot be told apart by eye, so the number on the circular scale cannot be raised without limit. The smaller the least count, the finer the reading.

2. Zero error and the reading

Before measuring, bring the faces A and B together and see whether both the linear and the circular scale read zero. If they do, the instrument has no zero error. If not, that leftover reading is the instrument's zero error.

Before the formula, pick up the gauge. Turn the thimble until the screw meets the wire, then read the two scales.

The reading is d = a + b×c, where a is the linear-scale reading, b is the circular-scale reading in divisions and c is the least count.

With c = 0.001 cm, a = 1.1 cm and b = 32 divisions, d = 1.1 + 32 × 0.001 = 1.132 cm.

Example 1.8(1): 100 divisions, pitch 1 mm, instrument error −0.02 mm; reading 2 mm and 28 divisions. L.C. = 0.01 mm, so d′ = 2 + 28 × 0.01 = 2.28 mm. The error is negative, so the correction is added: 2.28 + 0.02 = 2.30 mm = 0.230 cm. With a positive error it goes the other way: subtract. Run all three cases below.

3. The spherometer and the radius of curvature

A spherometer finds the radius of curvature of a spherical surface. The surface is part of some sphere, and the radius of that sphere is the radius of curvature.

The formula is R = d²/(6h) + h/2. Here d is the mean distance between the three legs, and h is the height or depth of the curved surface measured from the plane of the legs.

The least count is K = pitch / number of divisions on the circular scale. In one of the book's examples the smallest linear division is 0.01 cm with 100 divisions, so K = 10⁻⁴ cm. In another the pitch is 1 mm with 100 divisions, so K = 0.001 cm. Each is right under its own premise.

Reading = main-scale reading + circular-scale reading × least count.

Before ordering the steps, set the spherometer down yourself: on the flat glass first, then on the curved surface.

Take the reading X on a plane glass plate and Y on the curved surface. With the instrument error corrected, the two readings become X′ and Y′, and h = Y′ − X′. Measure the three leg-to-leg distances on a metre scale and average them to get d. Put the steps in order.

Precautions: turn the screw head the same way every time to avoid backlash, keep the glass plate clean, and check carefully that the screw tip is touching the surface.

Now the book's numbers. With d = (6.3 + 6.5 + 6.4)/3 = 6.4 cm and h = 4.96 cm, R = 6.4²/(6 × 4.96) + 4.96/2 = 1.376 + 2.48 = 3.86 cm. Creative question 2: d = 8 cm and h = 1.1 cm give R = 64/6.6 + 0.55 = 10.25 cm. Try your own numbers too.

If the same surface is concave instead of convex, the radius of curvature R does not change. Only the sign of h becomes negative.

4. The balance and the oscillation method

The balance finds mass by the oscillation method. Raise the beam and the pointer swings, then slowly settles. With both pans empty the rest point is X. With the body and a weight W₁ it is Y. With W₁ + 10 mg it is Z.

The pointer keeps swinging, so how do you find where it rests? Average three successive left turning points and two right ones, call the means x and y. The rest point is then ½(x + y).

If 10 mg moves the rest point by (Z − Y) divisions, one division stands for 10/(Z − Y) mg = 1/((Z − Y) × 100) g. That is how sensitive the balance is.

Precautions: make the pillar vertical and the base level. Lower and arrest the beam in the stirrup before placing or removing weights. Close the glass case before raising the beam. Read without parallax.

5. Know the instrument

Now match each instrument, by its job, to its least count.

Measured lengths often lead on to volumes, so keep these formulas close.

6. Check yourself

Three questions in the exam's style. Start with the first.

For the second, recall the formula.

Take a moment over the last one.

The whole note on seven cards.

Source: Physics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Physical World and Measurement”, section 1.14.