Physics · Class 9-10 · Chapter 1 · Physical quantities and measurement
The screw gauge, the balance and the stop watch.
A slide calipers stops at 0.1 of a millimetre. The diameter of a wire, or the thickness of a sheet of paper, is smaller than that. For those you need an instrument built on an ordinary screw, and the idea behind it is surprisingly simple.
1. The idea in a screw
Turn a screw through one full revolution and it advances a fixed distance along its thread. Notice what that means: the turn is large and the advance is small. One turn is a big movement of your hand, and the screw has gone forward by a single millimetre.
And there is the trick. Divide that one turn into a hundred parts, and turning through one part advances the screw by one hundredth of a millimetre. Dividing up the turn is what lets you measure a very small length, and that is the whole of the screw gauge. The instrument is also called a micrometer.
The names of the parts are unfamiliar, but they are the names your practical notebook wants. Tap the six points on the drawing and learn them.
spot themtap a dot on the drawing
The parts of a screw gauge
The frame. The stiff U shaped piece that holds everything else. Bend it and every reading the instrument ever gives is wrong.
The anvil. The fixed face. The object rests against it, and it never moves.
The spindle. The moving screw. Turn the thimble and it comes forward until the object is held against the anvil.
The sleeve and its linear scale. The barrel the spindle turns inside, marked off in millimetres. The further the thimble travels, the more marks come into view, and that is where the whole millimetres are read.
The thimble and its circular scale. The rim of the thimble usually carries a hundred divisions. Turn it by one and the spindle moves just 0.01 mm.
The ratchet. This is the knob you close the gap with. Past a set pressure it slips and clicks instead of tightening, so everybody measures at the same pressure and gets the same number.
All six. Notice there are two scales, one straight and one that turns. A reading needs both of them.
The drawing is a model, not a scale drawing of a real instrument.
2. Pitch and least count
The distance a screw advances along its own axis in one full turn is called its pitch. On an ordinary screw gauge the pitch is 1 mm, on some it is 0.5 mm. Judging one turn by eye is hard, so in practice you turn it ten times, read the total travel off the linear scale, and divide by ten.
pitch = total travel ÷ number of turns
Now the definition that matters. The distance the screw moves when the circular scale is turned by a single division is the least count of the instrument. It is the finest step the instrument can resolve.
LC = pitch ÷ number of circular divisions
With a 1 mm pitch and 100 divisions, LC = 1 ÷ 100 = 0.01 mm, which is 0.001 cm. The vernier constant of a slide calipers was 0.01 cm, so a screw gauge is ten times finer. That is why the diameter of a wire or the thickness of a thin sheet has to be measured with one.
check yourselfpick one answer
Find the least count
A screw gauge has a pitch of 1 mm and fifty divisions on its circular scale. What is its least count?
Right. LC = pitch ÷ divisions = 1 ÷ 50 = 0.02 mm, which is 0.002 cm.
0.01 mm would be right with a hundred divisions. That is the commonest instrument, so the number gets written down without reading the question. Use the number you were given.
Nothing in the question gives 0.5 mm. 1 ÷ 50 is not 0.5; this is what you get by dividing by 2 instead.
This multiplied where it should have divided. A least count is always smaller than the pitch, never larger.
3. Turn it and watch
The reading formula splits in two, exactly as it did for the slide calipers: the whole millimetres off the linear scale, the fraction off the circular one.
d = L + (C × LC)
Drag the handle below. On top is the linear scale on the sleeve, and every hundred divisions of turn brings out exactly one more millimetre mark. Below it is the circular scale, and the division that arrives at the index is C.
handledrag to turn it
Turn the screw gauge
The drawing is a model, not a scale drawing of a real instrument.
A worked example. The linear scale reads 3 mm, the circular division is 25, and LC = 0.01 mm. So d = 3 + (25 × 0.01) = 3.25 mm. Notice that three turns means three millimetres of travel, because the pitch is one millimetre.
An exam gives you these numbers, and usually a zero error with them. Change the boxes and watch the three steps stay separate: the least count first, then the observed reading, then the correction.
calculatorchange a number
Least count, and thickness
Here is how to read the sign of a zero error. Close the anvil on the spindle: if the zero of the circular scale sits below the line on the sleeve, the error is positive; if it sits above, it is negative. There is a second fault called backlash error: once the threads of the screw and its nut have worn, reversing the direction of turn leaves the thimble turning while the spindle stays put. There is one remedy, which is to always take a reading turning the same way.
check yourselfpick one answer
The true diameter of the wire
The linear scale reads 2 mm, the circular division is 45, and the least count is 0.01 mm. The instrument has a zero error of +5 divisions. What is the true diameter?
Right. The observed reading is 2 + (45 × 0.01) = 2.45 mm. The error is positive, +5 divisions or +0.05 mm, so the instrument reads 0.05 mm high every time. Subtract it: 2.45 − 0.05 = 2.40 mm.
This added the error. A positive error means the instrument reads high, so it comes off rather than on. The correction runs against the sign.
That is the observed reading, not the true diameter. The zero error correction has not been applied.
This treated 5 divisions as 0.5 mm. Five divisions is 5 × 0.01 = 0.05 mm, ten times less.
4. How fine, and how that shows in the reading
Everyone writes that a screw gauge is finer. Fewer people ask how much finer, or how that difference shows up in the number. Measure the same wire with both and look.
before and afterdrag the handle
0.1 mm against 0.01 mm
slide calipers
screw gauge
One wire, two instruments. The calipers can say the wire is 0.8 mm, but it does not know the next digit, so its band is wide. The screw gauge says 0.84 mm, and its band is ten times narrower. The finer the instrument, the more digits you are entitled to write. Writing more digits does not make a reading finer.
The drawing is a model, not a scale drawing.
Out of this comes a working rule for choosing an instrument, one that is rarely written down as a rule: the constant of the instrument should be at least ten times smaller than the thing you are measuring. Tap the ten jobs below and see which instrument each one needs.
which instrumenttap a job
What would you measure it with?
A measuring tape. Nobody needs a millimetre here, and you cannot lay a ruler along a hundred metres anyway.
A screw gauge. A page is about 0.1 mm, and 0.1 mm is the whole resolution of a calipers, so it cannot do it. The trick: measure a hundred pages together and divide by a hundred.
A screw gauge. A wire is usually under half a millimetre across, so nothing coarser than 0.01 mm will do.
A slide calipers. A few centimetres across, and 0.01 cm is plenty of resolution.
A slide calipers, using the small upper jaws. A screw gauge cannot measure an inside diameter at all; its faces only close inward.
A slide calipers, using the depth rod at the back. That is the third job a calipers can do and a screw gauge cannot.
A common balance or an electronic one. This is not a job for a length instrument at all.
A stop watch. But not one swing: time twenty of them and divide by twenty. The reason comes in a moment.
A metre scale. One millimetre is fine enough here, so an ordinary ruler does the job.
A screw gauge. A hair is roughly 0.07 mm, which is seven divisions of the circular scale and no more.
All ten. Did the rule show up? The constant of the instrument wants to be about a tenth of the thing you are measuring, or smaller.
5. Mass and time
The familiar instrument for mass is the common balance, which is really a lever with a pan on each side. The object goes on one, standard masses on the other, and equal tilt means equal mass. One thing is worth noticing: a balance measures mass, not weight. Gravity pulls equally on both pans, so the same balance would give the same reading on the moon. A sensitive balance reaches milligrams, and an electronic one goes finer still.
For time there is the stop watch, with a least count of 0.1 s on a mechanical one and 0.01 s on a digital. But however fine the instrument, your hand is not. Press the buttons and hear where the problem is.
listenpress one
Why twenty swings, not one
Here is the arithmetic. Starting and stopping a watch by hand carries about 0.2 s of reaction time. If one swing takes 1 s, that is a 20% error. Over twenty swings, about 20 s, the same 0.2 s divided by twenty comes to 0.01 s per swing, which is 1%. Count more and divide, and the error gets divided too.
check yourselfpick one answer
Why twenty swings
Why time twenty swings of a pendulum and divide by twenty, instead of timing one?
Right. Starting and stopping by hand leaves about 0.2 s of error. On one swing that is about 20%, but spread over twenty and then divided it comes to 0.01 s a swing.
It can: a stop watch resolves 0.1 s and one swing is about 1 s. The problem is not the instrument, it is the hand.
The period does not depend on how many swings you count, and that steadiness is the whole point of a pendulum. It depends on the length and on g.
A pendulum does not stop at twenty; it dies away much later. The twenty comes from the arithmetic of the error, not from the pendulum.