Chapter 1: Measurements And Experimentation
On 11 December 1998, a spacecraft called the Mars Climate Orbiter left Earth. It was built to circle Mars and study its thin atmosphere and its weather. After a journey of more than nine months, it reached Mars on 23 September 1999 — and was never heard from again.
The investigation that followed did not find a cracked engine or a failed computer. It found a unit. One team had written its software to report the small pushes of the spacecraft’s thrusters in a British engineering unit, the pound-force second. The navigation team receiving those numbers assumed they were in the metric unit, the newton second. The two teams exchanged numbers faithfully for months. Nobody exchanged the unit.
Because one newton second is a much smaller push than one pound-force second, every correction was quietly wrong. The path drifted. Instead of settling into orbit about 226 kilometres above the surface, the spacecraft came in far too low, met the atmosphere, and was destroyed.
Two hundred years of careful science, a working spacecraft, and a mission worth many crores were lost to a number written without its unit.
That is why this chapter begins where physics itself begins: with what a measurement really is, which units the world has agreed to share, and how an instrument in your own hands is read correctly, down to its last honest decimal place.
Diagram
Fig. 1.0-1 — A Measurement Is A Number And A Unit Together Two-part illustration. Left panel: a spacecraft on a curved dashed path approaching a planet labelled Mars, with two arrows leaving a thruster — one short arrow labelled ‘1 newton second’ and one much longer arrow labelled ‘1 pound-force second’ — showing that the same figure ‘1’ means two different pushes; mark the intended high orbit path and the too-low actual path entering the atmosphere. Right panel: a simple balance sheet style box showing the words ‘numerical value’ and ‘unit’ joined by a multiplication sign, with the example ‘5 kg’ underneath, the 5 labelled ‘how many times’ and the kg labelled ‘the standard chosen’.
1.1 Need Of A Unit For Measurement
Physics is an experimental science. Almost every idea in it — how fast, how heavy, how hot, how far — has to be settled by measuring something and writing down what you found.
A physical quantity is anything about an object or an event that can be measured, such as its length, its mass or the time it takes. To measure one, you first choose a fixed quantity of the same kind to act as a standard. You then find how many times that standard is contained in the quantity you are measuring.
Suppose you lay a metre stick along a rope, end to end, and it fits exactly four times. You have compared the rope with a standard length and found the answer to be four of them. This act of comparison is what measuring means, and it is shown in Fig. 1.1-1.
Diagram
Fig. 1.1-1 — Measuring Is Comparing With A Standard A long rope drawn horizontally. Below it, four identical sticks each labelled ‘1 metre’ laid end to end so that together they exactly span the rope. Label the rope ‘quantity to be measured’ and the row of sticks ‘the standard, taken 4 times’. To the right, write the result ‘length of rope = 4 metre’ with ‘4’ labelled ‘numerical value’ and ‘metre’ labelled ‘unit’.
The standard quantity chosen for this comparison is called the unit. Quantities of different nature need units of different nature — you cannot measure a mass with a standard length, because they are not comparable at all.
1.1.1 Expressing The Magnitude Of A Physical Quantity
The result of a measurement is never a bare number. It always has two parts joined together:
- the unit in which the quantity has been measured, and - the numerical value, which tells how many times that unit is contained in the quantity.
Written as a relation, this is:
\[\\text{physical quantity} = (\\text{numerical value}) \\times (\\text{unit})\]
If the length of a piece of cloth is \(10\\ \\text{metre}\), the unit chosen is the metre, and the number \(10\) says that this unit fits ten times into the cloth. If a bag of sugar has a mass of \(5\\ \\text{kilogram}\), the unit chosen is the kilogram, and it is contained five times in that mass.
Notice what follows from this. Change the unit and the number must change too, because the same quantity is now being counted in smaller or larger steps. The quantity itself does not change at all.
1.1.2 Choice Of A Unit
A unit is meant to be used by everyone, everywhere, and to mean exactly the same thing each time. A unit chosen for measuring a physical quantity should therefore have these properties:
- It should be of convenient size, so that ordinary measurements give neither enormous nor tiny numbers.
- It should be possible to define it without any ambiguity.
- It should be reproducible, so that a copy of the standard can be made wherever it is needed.
- Its value should not change with space and time — it must remain the same everywhere and always.
The last three conditions are the essential ones for a unit to be accepted internationally. Convenience only makes a unit pleasant to work with; the other three are what make two laboratories on opposite sides of the world agree.
1.1.3 Kinds Of Units
All units fall into two kinds: fundamental (or basic) units, and derived units.
1.1.3.1 Fundamental Or Basic Units
Some quantities are of an entirely different nature from one another. Mass, length, time, temperature, electric current and amount of substance cannot be obtained from one another, and the unit of one cannot be built out of the unit of another. Their units are taken as the starting point of the whole system.
1.1.3.2 Derived Units
Most quantities are not of this independent kind. Their units are built by combining fundamental units, following the way the quantity itself is defined.
Three examples show how the building works:
- Area is length multiplied by breadth, and both are lengths. Its unit is therefore \(\\text{length} \\times \\text{length}\), or \((\\text{length})^2\).
- Volume is length multiplied by breadth multiplied by height. Its unit is \((\\text{length})^3\).
- The speed of a moving body is found by dividing the distance covered by the time taken. Its unit is therefore the unit of length divided by the unit of time.
So the units of area, volume and speed are all derived units. Many more examples of derived units are collected later in the chapter, in section 1.6.
1.2 Systems Of Unit
In mechanics — the study of matter in motion — three quantities are treated as fundamental: length, mass and time. Everything else in mechanics is built from these three.
A system of units is a matched set of units chosen for these basic quantities. Over the years three such systems have been in use, and they differ only in which unit is picked for each quantity.
1.2.1 The C.G.S., F.P.S. And M.K.S. Systems
The three older systems are set side by side in Table 1.1. The names are simply the initials of the units they use.
These systems are no longer in use and are now only of historical importance. For the sake of uniformity, the scientific community has agreed to use the S.I. system of units, which is an enlarged and modified version of the metric system.
1.2.2 The S.I. System And Its Fundamental Units
The full name of this system is Système Internationale d’Unités, shortened everywhere to the S.I. system. It was established by the General Conference of Weights and Measures, which recommended that besides the units of length, mass and time, the units of temperature, luminous intensity, electric current and amount of substance also be taken as fundamental units, and the units of angle and solid angle be taken as supplementary fundamental units.
There are therefore seven fundamental units and two supplementary units in all. Their quantities, units and standard symbols are listed in Table 1.2.
| Quantity | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | K |
| Luminous intensity | candela | cd |
| Electric current | ampere | A |
| Amount of substance | mole | mol |
| Angle | radian | rad |
| Solid angle | steradian | sr |
Table 1.2 — Fundamental quantities, units and symbols in the S.I. system. The first seven are the fundamental units; the last two, the radian and the steradian, are the supplementary units.
The mole needs one word of explanation, since it is the only one of the seven that counts things rather than measures them. One mole is the amount of substance that contains as many constitutive entities — electrons, photons, ions, atoms or molecules — as there are atoms in \(12\\ \\text{g}\) of carbon-12. This number is approximately \(6.022 \\times 10^{23}\\ \\text{mole}^{-1}\) and is known as the Avogadro constant.
1.2.3 Use Of A Prefix With A Unit
Some measurements are enormous and some are minute. Writing them out in the plain unit gives long strings of zeros, so a short word called a prefix is attached to the front of the unit instead. Each prefix stands for a fixed power of ten, as Fig. 1.2-1 shows.
Diagram
Fig. 1.2-1 — A Prefix Stands For A Power Of Ten A horizontal number line marked with powers of ten from 10⁻¹² at the far left to 10¹² at the far right, with 10⁰ marked at the centre. Above the line, at the correct positions, place the prefix names and symbols: pico (p) at 10⁻¹², nano (n) at 10⁻⁹, micro (µ) at 10⁻⁶, milli (m) at 10⁻³, centi (c) at 10⁻², deci (d) at 10⁻¹, deca (da) at 10¹, hecto (h) at 10², kilo (k) at 10³, mega (M) at 10⁶, giga (G) at 10⁹, tera (T) at 10¹². Below the line, mark an arrow to the left labelled ‘smaller measurements’ and an arrow to the right labelled ‘larger measurements’.
For expressing large measurements we use deca, hecto, kilo and so on as prefixes with the units. Their symbols and meanings are given in Table 1.3.
| Prefix | Symbol | Meaning |
|---|---|---|
| deca | da | \(10^{1}\) |
| hecto | h | \(10^{2}\) |
| kilo | k | \(10^{3}\) |
| mega | M | \(10^{6}\) |
| giga | G | \(10^{9}\) |
| tera | T | \(10^{12}\) |
| peta | P | \(10^{15}\) |
| exa | E | \(10^{18}\) |
| zetta | Z | \(10^{21}\) |
| yotta | Y | \(10^{24}\) |
Table 1.3 — Some prefixes used for big measurements.
Small measurements are expressed by using the prefixes deci, centi, milli, micro and so on with the units. Their symbols and meanings are given in Table 1.4.
| Prefix | Symbol | Meaning |
|---|---|---|
| deci | d | \(10^{-1}\) |
| centi | c | \(10^{-2}\) |
| milli | m | \(10^{-3}\) |
| micro | \(\\mu\) | \(10^{-6}\) |
| nano | n | \(10^{-9}\) |
| pico | p | \(10^{-12}\) |
| femto | f | \(10^{-15}\) |
| atto | a | \(10^{-18}\) |
| zepto | z | \(10^{-21}\) |
| yocto | y | \(10^{-24}\) |
Table 1.4 — Some prefixes used for small measurements.
To read such a quantity, simply replace the prefix by the power of ten it stands for. Thus \(2.5\\ \\text{GHz}\) means \(2.5 \\times 10^{9}\\ \\text{Hz}\), \(5.0\\ \\text{pF}\) means \(5.0 \\times 10^{-12}\\ \\text{F}\), \(5.0\\ \\text{M}\\Omega\) means \(5.0 \\times 10^{6}\\ \\Omega\), and \(2.0\\ \\text{ms}\) means \(2.0 \\times 10^{-3}\\ \\text{s}\).
1.3 Units Of Length
The lengths a physicist deals with run from the width of a nucleus to the gap between two stars. No single unit is comfortable across that whole range, so the metre is used together with a family of smaller and bigger units, all defined in terms of it. Fig. 1.3-1 gives a sense of the range involved.
Diagram
Fig. 1.3-1 — The Range Of Lengths Physicists Measure A vertical ladder scale marked in powers of ten of a metre, from 10⁻¹⁵ m at the bottom to 10¹⁶ m at the top. Mark rungs with the unit name and a familiar object at each: 10⁻¹⁵ m ‘fermi — size of a nucleus’; 10⁻¹⁰ m ‘angstrom — size of an atom’; 10⁻⁹ m ‘nanometre — wavelength of light’; 10⁻⁶ m ‘micron — a bacterium’; 10⁻³ m ‘millimetre — thickness of a coin’; 10⁰ m ‘metre — height of a desk’; 10³ m ‘kilometre — a long walk’; 10¹¹ m ‘astronomical unit — earth to sun’; 10¹⁶ m ‘light year and parsec — earth to a star’. Label the axis ‘length in metre’.
1.3.1 The S.I. Unit Of Length And How It Is Defined
The S.I. unit of length is the metre, symbol m. What has changed over the years is not the size of the metre but the way it is pinned down.
A metre was originally defined in 1889 as the distance between two marks drawn on a platinum-iridium rod — an alloy of \(90%\) platinum and \(10%\) iridium — kept at \(0^\\circ\\text{C}\) at the International Bureau of Weights and Measures at Sèvres near Paris.
Later, in 1960, the metre was re-defined as \(1{,}650{,}763.73\) times the wavelength of a specified orange-red spectral line in the emission spectrum of krypton-86. It is also defined as \(1{,}553{,}164.1\) times the wavelength of the red line in the emission spectrum of cadmium.
1.3.2 Sub Units Of The Metre
For measuring small lengths the metre is too big a unit. The sub units in common use are the centimetre, the millimetre, the micron and the nanometre.
- Centimetre (cm) — one centimetre is one-hundredth part of a metre, so \(1\\ \\text{cm} = \\frac{1}{100}\\ \\text{m} = 10^{-2}\\ \\text{m}\).
- Millimetre (mm) — one millimetre is one-thousandth part of a metre, so \(1\\ \\text{mm} = \\frac{1}{1000}\\ \\text{m} = 10^{-3}\\ \\text{m} = \\frac{1}{10}\\ \\text{cm}\).
- Micron (\(\\mu\)) — it is one-millionth part of a metre. It is also called the micrometre, symbol \(\\mu\\text{m}\). Thus \(1\\ \\mu = 10^{-6}\\ \\text{m} = 10^{-4}\\ \\text{cm} = 10^{-3}\\ \\text{mm}\).
- Nanometre (nm) — it is one billionth part of a metre, that is \(1\\ \\text{nm} = 10^{-9}\\ \\text{m}\).
1.3.3 Multiple Unit Of The Metre
For measuring large lengths or distances the metre is too small a unit. The multiple unit in common use is the kilometre.
Kilometre (km) — one kilometre is the one-thousand multiple of a metre, that is \(1\\ \\text{km} = 1000\\ \\text{m}\), or \(10^{3}\\ \\text{m}\).
1.3.4 Non-Metric Units Of Length
Beyond this range, two families of units are used which are not simple multiples of ten of the metre. One is for the distances between heavenly bodies, the other for the sizes of atoms and nuclei.
1.3.4.1 Bigger Units — Astronomical Unit, Light Year And Parsec
For the distance between two heavenly bodies even the kilometre is far too small. Three units are used for this purpose.
Astronomical unit (A.U.) — one astronomical unit is equal to the mean distance between the earth and the sun:
\[1\\ \\text{A.U.} = 1.496 \\times 10^{11}\\ \\text{metre}\]
Light year (ly) — a light year is the distance travelled by light in vacuum in one year. Although its name sounds like a time, it is a distance, because it is found by multiplying a speed by a time:
\[1\\ \\text{light year} = \\text{speed of light} \\times \\text{time of one year}\] \[= (3 \\times 10^{8}\\ \\text{m s}^{-1}) \\times (365 \\times 24 \\times 60 \\times 60\\ \\text{s}) = 9.46 \\times 10^{15}\\ \\text{m} = 9.46 \\times 10^{12}\\ \\text{km}\]
The distance of stars from the earth is generally expressed in light years. Its smaller relatives are also used:
\[1\\ \\text{light minute} = (3 \\times 10^{8}\\ \\text{m s}^{-1}) \\times 60\\ \\text{s} = 1.8 \\times 10^{10}\\ \\text{m}\] \[1\\ \\text{light second} = (3 \\times 10^{8}\\ \\text{m s}^{-1}) \\times 1\\ \\text{s} = 3 \\times 10^{8}\\ \\text{m}\]
Parsec — one parsec is the distance from where the semi-major axis of the orbit of the earth, which is \(1\\ \\text{A.U.}\), subtends an angle of one second of arc. The name itself is made from two words, parallax and arc-second. The arrangement is drawn in Fig. 1.3-2.
Diagram
Fig. 1.3-2 — How One Parsec Is Defined A long horizontal line from a point marked S on the right to a point on the left; at the left end draw a thin tall triangle whose short vertical side is labelled ‘1 A.U. = 1.496 × 10¹¹ m’ and is marked ‘semi-major axis of the earth’s orbit’. The angle at S between the two long sides is marked as 1″ (one second of arc), drawn exaggerated for clarity with a note ‘angle shown much larger than actual’. Label the long horizontal distance ‘1 parsec = 3.08 × 10¹⁶ m’.
An angle is turned into a length by using the fact that for a small angle, the arc length equals the radius multiplied by the angle in radians. So:
\[1\\ \\text{parsec} = \\frac{\\text{arc length}}{\\text{angle in radians}} = \\frac{1.496 \\times 10^{11}\\ \\text{m}}{(1/3600) \\times (\\pi/180)} = 3.08 \\times 10^{16}\\ \\text{m}\]
Here the angle of one second has been converted to radians in two steps: dividing by \(3600\) turns seconds of arc into degrees, and multiplying by \(\\pi/180\) turns degrees into radians. Comparing the parsec with the light year:
\[1\\ \\text{parsec} = \\frac{3.08 \\times 10^{16}}{9.46 \\times 10^{15}}\\ \\text{ly} = 3.26\\ \\text{light year}\]
1.3.4.2 Smaller Units — Angstrom And Fermi
To express the wavelength of light, the size of and separation between two molecules or atoms, and the radius of the orbit of an electron, a small unit called the angstrom (\(\\text{Å}\)) is used. The size of a nucleus is expressed by a still smaller unit called the fermi (f).
Angstrom (Å) — it is the \(10^{-10}\)th part of a metre:
\[1\\ \\text{Å} = 10^{-10}\\ \\text{m} = 10^{-8}\\ \\text{cm} = 10^{-1}\\ \\text{nm}\]
From this it follows that \(1\\ \\mu = 10{,}000\\ \\text{Å}\) and \(1\\ \\text{nm} = 10\\ \\text{Å}\).
Nowadays the angstrom is outdated and the nanometre is preferred over it. The wavelength of light, and inter-atomic and inter-molecular separations, are now commonly expressed in nm.
Fermi (f) — it is the \(10^{-15}\)th part of a metre, that is \(1\\ \\text{f} = 10^{-15}\\ \\text{m}\).
The commonly used smaller and bigger units of length are collected in Table 1.5.
| Smaller units | Value in metre | Bigger units | Value in metre |
|---|---|---|---|
| cm | \(10^{-2}\\ \\text{m}\) | km | \(10^{3}\\ \\text{m}\) |
| mm | \(10^{-3}\\ \\text{m}\) | A.U. | \(1.496 \\times 10^{11}\\ \\text{m}\) |
| \(\\mu\) (or \(\\mu\\text{m}\)) | \(10^{-6}\\ \\text{m}\) | ly | \(9.46 \\times 10^{15}\\ \\text{m}\) |
| nm | \(10^{-9}\\ \\text{m}\) | parsec | \(3.08 \\times 10^{16}\\ \\text{m}\) |
| Å | \(10^{-10}\\ \\text{m}\) | ||
| f | \(10^{-15}\\ \\text{m}\) |
Table 1.5 — Smaller and bigger units of length.
1.4 Units Of Mass
Mass tells how much matter an object contains. Like length, it is measured over an enormous range — from a single atom to the sun — so the kilogram is used alongside a family of smaller and bigger units. Fig. 1.4-1 sets out that range.
Diagram
Fig. 1.4-1 — The Range Of Masses Physicists Measure A vertical ladder scale marked in powers of ten of a kilogram, from 10⁻²⁷ kg at the bottom to 10³⁰ kg at the top. Mark rungs with the unit and a familiar object: 10⁻²⁷ kg ‘one atomic mass unit — a single proton’; 10⁻⁶ kg ‘milligram — a grain of sand’; 10⁻³ kg ‘gram — a paper clip’; 10⁰ kg ‘kilogram — a litre of water’; 10² kg ‘quintal — a sack of grain’; 10³ kg ‘metric tonne — a small car’; 10³⁰ kg ‘solar mass — the sun’. Label the axis ‘mass in kilogram’.
1.4.1 The S.I. Unit Of Mass And How It Is Defined
The S.I. unit of mass is the kilogram, symbol kg. A convenient everyday check on its size is that the mass of \(1\\ \\text{litre}\), that is \(1000\\ \\text{ml}\), of water at \(4^\\circ\\text{C}\) is also taken as \(1\\ \\text{kilogram}\).
In 1889, one kilogram was defined as the mass of a cylindrical piece of platinum-iridium alloy kept at the International Bureau of Weights and Measures at Sèvres near Paris.
That standard had the same weakness as the metre bar: it was one object in one place. The definition in force since 20th May 2019 therefore defines mass in terms of a fundamental value of nature called the Planck constant, whose value is \(6.62607015 \\times 10^{-34}\\ \\text{kg m}^{2},\\text{s}^{-1}\). The instrument that puts this definition to work is the Kibble balance, which measures the mass of an object using a precisely measured electromagnetic force.
1.4.2 Sub Units Of The Kilogram
For measuring small masses the kilogram is too big a unit. The smaller units in common use are the gram and the milligram.
- Gram (g) — one gram is the one-thousandth part of a kilogram, so \(1\\ \\text{g} = \\frac{1}{1000}\\ \\text{kg} = 10^{-3}\\ \\text{kg}\), or \(1\\ \\text{kg} = 1000\\ \\text{g}\).
- Milligram (mg) — one milligram is one-millionth part of a kilogram, or one-thousandth part of a gram, so \(1\\ \\text{mg} = 10^{-6}\\ \\text{kg}\) and \(1\\ \\text{mg} = 10^{-3}\\ \\text{g}\).
1.4.3 Multiple Units Of The Kilogram
The bigger units of mass used in daily life are the quintal and the metric tonne.
- Quintal — it is one hundred times a kilogram, that is \(1\\ \\text{quintal} = 100\\ \\text{kg}\).
- Metric tonne — it is one thousand times a kilogram, that is \(1\\ \\text{metric tonne} = 1000\\ \\text{kg} = 10\\ \\text{quintal}\).
1.4.4 Non-Metric Units Of Mass
The mass of atomic particles such as the proton, the neutron and the electron is expressed in a unit called the atomic mass unit, symbol a.m.u., or the unified atomic mass unit, symbol u.
Its value in kilogram follows from the fact that \(6.02 \\times 10^{23}\) atoms of carbon-12 together have a mass of \(12\\ \\text{g}\), which is \(12 \\times 10^{-3}\\ \\text{kg}\). The mass of one such atom is that total divided by the number of atoms, and one a.m.u. is one-twelfth of it:
\[1\\ \\text{a.m.u.} = \\frac{1}{12} \\times \\frac{12 \\times 10^{-3}\\ \\text{kg}}{6.02 \\times 10^{23}} = 1.66 \\times 10^{-27}\\ \\text{kg}\]
At the other extreme, the mass of large heavenly bodies is measured in terms of the solar mass, where one solar mass is the mass of the sun:
\[1\\ \\text{solar mass} = 2 \\times 10^{30}\\ \\text{kg}\]
The commonly used smaller and bigger units of mass are collected in Table 1.6.
| Smaller units | Value in kg | Bigger units | Value in kg |
|---|---|---|---|
| g | \(10^{-3}\\ \\text{kg}\) | quintal | \(100\\ \\text{kg}\) |
| mg | \(10^{-6}\\ \\text{kg}\) | metric tonne | \(1000\\ \\text{kg}\) |
| u (or a.m.u.) | \(1.66 \\times 10^{-27}\\ \\text{kg}\) | solar mass | \(2 \\times 10^{30}\\ \\text{kg}\) |
Table 1.6 — Smaller and bigger units of mass.
1.5 Units Of Time
Time is measured by counting something that repeats itself faithfully — the turning of the earth, the swing of a pendulum, the vibration of an atom. Every unit of time in this section is built on one of these repetitions.
1.5.1 The S.I. Unit Of Time And How It Is Defined
The S.I. unit of time is the second, symbol s. The oldest definition took the earth itself as the clock.
One solar day is the time taken by the earth to complete one rotation on its own axis, as shown in Fig. 1.5-1. A mean solar day is the average of these over a year, since individual days are not exactly equal. Dividing that day into \(24\) hours, each of \(60\) minutes, each of \(60\) seconds, gives \(24 \\times 60 \\times 60 = 86400\) parts, which is where the number in the definition comes from.
Diagram
Fig. 1.5-1 — One Solar Day Is One Rotation Of The Earth A globe drawn twice, side by side, with its axis tilted and marked with a curved arrow showing the direction of rotation. On the first globe mark a small flag at a point facing the sun, with rays from a sun drawn at the left. On the second globe show the same flag back in the same position after a full turn. Label the arrow between the two globes ‘one complete rotation = one solar day = 86400 s’. Do not mark any national boundaries on the globe.
For many years this definition remained in use. But the mean solar day varies slowly over the years, so in 1956 scientists agreed to fix the second against one particular year instead. They took the year 1900 as the standard, whose length was \(365.2422\) days:
\[1\\ \\text{year }1900 = 365.2422\\ \\text{days} = 365.2422 \\times 86400\\ \\text{s} = 31556926.08\\ \\text{s}\]
Hence one second was defined as the \(\\dfrac{1}{31{,}556{,}926.08}\)th part of the year 1900.
Even a chosen year is an awkward standard, because nobody can go back and check it. In 1964 the second was therefore defined in terms of an energy change inside an atom, which any laboratory can set up for itself.
1.5.2 Smaller Units Of Time
Many events in physics are far shorter than a second. The common smaller units are the millisecond (ms), the microsecond (\(\\mu\\text{s}\)), the shake and the nanosecond (ns):
\[1\\ \\text{ms} = 10^{-3}\\ \\text{s}, \\qquad 1\\ \\mu\\text{s} = 10^{-6}\\ \\text{s}, \\qquad 1\\ \\text{shake} = 10^{-8}\\ \\text{s}, \\qquad 1\\ \\text{ns} = 10^{-9}\\ \\text{s}\]
1.5.3 Bigger Units Of Time
For longer intervals the second is too small, so we use units built up from the day and the year.
- Minute (min) — one minute is the duration of \(60\) second, that is \(1\\ \\text{min} = 60\\ \\text{s}\).
- Hour (h) — one hour is the duration of \(60\) minutes, so \(1\\ \\text{h} = 60\\ \\text{min} = 60 \\times 60\\ \\text{s} = 3600\\ \\text{s}\).
- Day — the time taken by the earth to rotate once on its own axis is called a day. One day is divided into \(24\) hours, so \(1\\ \\text{day} = 24\\ \\text{h} = 24 \\times 60\\ \\text{min} = 1440\\ \\text{min} = 24 \\times 60 \\times 60\\ \\text{s} = 86400\\ \\text{s}\).
Months And Years
- Lunar month — the western or Gregorian calendar is based on the period of revolution of the earth around the sun, but our Hindu (Vikram and Shak) and Muslim (Hizri) calendars are based on the phases of the moon as seen from the earth. In these calendars one month is the time of one lunar cycle, which is nearly \(29.5\) days. Twelve lunar months thus come to \(354.37\) days, so one year of these calendars is short of a Gregorian year by nearly \(10\) days.
- Month — the Gregorian calendar has January, March, May, July, August, October and December each of \(31\) days; April, June, September and November each of \(30\) days; and February of \(28\) days, or \(29\) days in a leap year. To an approximation, a month is taken to be of \(30\) days and a year of \(12\) months, or \(365\) days.
- Year (yr) — one year is defined as the time in which the earth completes one revolution around the sun. This period is nearly \(365.256\) days when the distant stars are taken as the reference point for the earth’s position, and a year measured this way is called a sidereal year. To an approximation, \(1\\ \\text{yr} = 365\\ \\text{days} = 365 \\times 86400\\ \\text{s} = 3.1536 \\times 10^{7}\\ \\text{s}\).
- Leap year — a leap year is the year in which the month of February is of \(29\) days, so that \(1\\ \\text{leap year} = 366\\ \\text{days}\).
Why Leap Years Exist
When the sun is taken as the reference point for position, the year comes to \(365.2422\) days; this is called a tropical year and it is the year used for the Gregorian calendar. Since the calendar year is counted as \(365\) days, an excess of \(0.2422\) day is left over every year. To compensate for it, the Gregorian calendar has been modified as follows.
Every fourth year — that is, a year divisible by \(4\) — has one extra day in February, which then has \(29\) days, and so it is a leap year. The years 1904, 1908, 2000, 2004, 2008, 2012 and 2016 were leap years, and 2020, 2024 and so on will also be leap years. There is one exception. The century years, such as 1800 and 1900, though divisible by \(4\), are not leap years; the year 2100 will not be a leap year. But century years which are divisible by \(400\) are leap years, so the year 2000 was a leap year and the year 2400 will also be one. The whole rule is set out in FC 1.1.
Flowchart
FC 1.1 — Deciding Whether A Year Is A Leap Year Take the year → Is it divisible by 4? → if No: not a leap year → if Yes: Is it a century year (divisible by 100)? → if No: leap year → if Yes: Is it divisible by 400? → if Yes: leap year → if No: not a leap year
- Decade — a decade is of \(10\) years, so \(1\\ \\text{decade} = 10\\ \\text{years} = 3.1536 \\times 10^{8}\\ \\text{s}\).
- Century — a century is of \(100\) years. In a century there are \(24\) years of \(366\) days and \(76\) years of \(365\) days. Thus \(1\\ \\text{century} = (24 \\times 366 + 76 \\times 365)\\ \\text{days} = 36524\\ \\text{days} = 3.16 \\times 10^{9}\\ \\text{s}\).
- Millennium — a millennium is of \(1000\) years, so \(1\\ \\text{millennium} = 3.16 \\times 10^{10}\\ \\text{s}\).
The commonly used bigger units of time are collected in Table 1.7.
| Bigger units | Value in second | Bigger units | Value in second |
|---|---|---|---|
| min | \(60\\ \\text{s}\) | year | \(3.1536 \\times 10^{7}\\ \\text{s}\) |
| h | \(3600\\ \\text{s}\) | decade | \(3.1536 \\times 10^{8}\\ \\text{s}\) |
| day | \(86400\\ \\text{s}\) | century | \(3.16 \\times 10^{9}\\ \\text{s}\) |
| month | \(2.592 \\times 10^{6}\\ \\text{s}\) | millennium | \(3.16 \\times 10^{10}\\ \\text{s}\) |
Table 1.7 — Bigger units of time.
1.6 Some Examples Of Derived Units
Apart from the seven fundamental quantities used in the S.I. system — length, mass, time, temperature, luminous intensity, electric current and amount of substance — the units of all other physical quantities are obtained in terms of the fundamental units. The units so obtained are called derived units.
The recipe is always the same. Write down how the quantity is defined, then replace each quantity in that definition by its own unit. Fig. 1.6-1 and FC 1.2 follow this route for the unit of force.
Flowchart
FC 1.2 — Building The Unit Of Force From Fundamental Units Unit of length (metre) and unit of time (second) → unit of speed (metre per second) → unit of acceleration (metre per second squared) → multiply by unit of mass (kilogram) → unit of force (kilogram metre per second squared, named the newton)
Diagram
Fig. 1.6-1 — A Derived Unit Is Assembled, Not Invented A simple assembly sketch. Three labelled blocks on the left — ‘kilogram (mass)’, ‘metre (length)’, ‘second (time)’ — with arrows feeding into a single box on the right labelled ‘kg m s⁻²’. Below that box write ‘called the newton, symbol N’. Above the arrows write the defining sentence ‘force = mass × acceleration’. Mark the metre and second blocks as feeding first into a small intermediate box labelled ‘acceleration, m s⁻²’.
A short list of derived quantities, with their definitions and units, is given in Table 1.8. Read each row from left to right: the definition tells you what to multiply or divide, and the derived unit is the result of doing exactly that to the fundamental units.
| Quantity | Definition | Derived unit | Symbol |
|---|---|---|---|
| Area | length \(\\times\) breadth | metre \(\\times\) metre | \(\\text{m}^{2}\) |
| Volume | length \(\\times\) breadth \(\\times\) height | metre \(\\times\) metre \(\\times\) metre | \(\\text{m}^{3}\) |
| Density | \(\\dfrac{\\text{mass}}{\\text{volume}}\) | \(\\dfrac{\\text{kilogram}}{(\\text{metre})^{3}}\) | \(\\text{kg m}^{-3}\) |
| Speed or velocity | \(\\dfrac{\\text{distance}}{\\text{time}}\) | \(\\dfrac{\\text{metre}}{\\text{second}}\) | \(\\text{m s}^{-1}\) |
| Acceleration | \(\\dfrac{\\text{velocity}}{\\text{time}}\) | \(\\dfrac{\\text{metre/second}}{\\text{second}}\) | \(\\text{m s}^{-2}\) |
| Force | mass \(\\times\) acceleration | kilogram \(\\times \\dfrac{\\text{metre}}{(\\text{second})^{2}}\), or newton | \(\\text{kg m s}^{-2}\) or N |
| Work or energy | force \(\\times\) displacement | kilogram \(\\times \\dfrac{\\text{metre}}{(\\text{second})^{2}} \\times\) metre, or joule | \(\\text{kg m}^{2},\\text{s}^{-2}\) or J |
| Momentum | mass \(\\times\) velocity | kilogram \(\\times \\dfrac{\\text{metre}}{\\text{second}}\), or newton \(\\times\) second | \(\\text{kg m s}^{-1}\) or N s |
| Moment of force or torque | force \(\\times\) distance | kilogram \(\\times \\dfrac{\\text{metre}}{(\\text{second})^{2}} \\times\) metre, or newton-metre | \(\\text{kg m}^{2},\\text{s}^{-2}\) or N m |
| Power | \(\\dfrac{\\text{work}}{\\text{time}}\) | joule per second, or watt | \(\\text{kg m}^{2},\\text{s}^{-3}\), or \(\\text{J s}^{-1}\) or W |
| Pressure | \(\\dfrac{\\text{force}}{\\text{area}}\) | newton per \((\\text{metre})^{2}\), or pascal | \(\\text{kg m}^{-1},\\text{s}^{-2}\), or \(\\text{N m}^{-2}\) or Pa |
| Frequency | \(\\dfrac{1}{\\text{time period}}\) | \(\\dfrac{1}{\\text{second}}\), or hertz | \(\\text{s}^{-1}\) or Hz |
| Electric charge | current \(\\times\) time | ampere \(\\times\) second, or coulomb | A s or C |
| Electric potential or electromotive force (e.m.f.) | \(\\dfrac{\\text{work}}{\\text{charge}}\) | joule per coulomb, or volt | \(\\text{kg m}^{2},\\text{A}^{-1},\\text{s}^{-3}\), or \(\\text{J C}^{-1}\) or V |
| Electrical resistance | \(\\dfrac{\\text{potential}}{\\text{current}}\) | volt per ampere, or ohm | \(\\text{kg m}^{2},\\text{A}^{-2},\\text{s}^{-3}\), or \(\\text{V A}^{-1}\) or \(\\Omega\) |
| Electrical power | potential \(\\times\) current | volt \(\\times\) ampere, or watt | V A or W |
Table 1.8 — Derived units of some physical quantities.
Units Named After Scientists
From the list above it can be noted that some derived units are complex when they are expressed in terms of fundamental units. Such derived units have been given special names after the name of the scientist who has contributed in that field — newton for force, joule for work or energy, watt for power, pascal for pressure, hertz for frequency, coulomb for electric charge, volt for electric potential or e.m.f., and ohm for electrical resistance.
The full name of such a unit is always written with a small first letter, while its symbol is written with its first letter in capital. So we write \(5\\ \\text{newton}\) but \(5\\ \\text{N}\), and \(2\\ \\text{joule}\) but \(2\\ \\text{J}\).
1.7 Guidelines For Writing The Units
A unit is a piece of information, and like any other piece of information it can be spoilt by careless writing. Conventionally, the following rules are observed while writing the unit of a physical quantity. Fig. 1.7-1 shows the parts of a correctly written quantity.
Diagram
Fig. 1.7-1 — The Parts Of A Correctly Written Physical Quantity A single large example written across the page: 2.5 kN. Draw three labelled arrows pointing to its parts — one to ‘2.5’ labelled ‘numerical value’; one to ‘k’ labelled ‘prefix, standing for 10³’; one to ‘N’ labelled ‘unit symbol, capital N because the newton is named after a scientist’. Mark the gap between 2.5 and kN with a small bracket labelled ‘always leave a space’. Below, show the same quantity written wrongly as ‘2.5 KNs.’ with the capital K, the added s and the full stop each struck through.
The symbol for a unit which is not named after a scientist is written in a small letter. For example, the symbol for metre is m, for second is s, and for kilogram is kg.
The symbol for a unit which is named after a scientist is written with the first letter of the name in capital. For example, N for newton, J for joule, W for watt, Pa for pascal, Hz for hertz, C for coulomb and V for volt. There is one exception. Although the litre, the unit of volume, is not named after a scientist, its symbol is sometimes written as ‘L’ instead of ‘l’, so as to avoid confusion between the letters ‘l’ and ‘I’.
The full name of the unit, whether or not it is named after a scientist, is always written with a lower initial letter. So the unit of mass is written as kilogram, not Kilogram; the unit of length as metre, not Metre; the unit of force as newton, not Newton; the unit of energy as joule, not Joule; and the unit of power as watt, not Watt.
A compound unit formed by multiplication of two or more units is written by putting a dot or a cross between the two symbols, or by leaving a space between them. For example, the unit of torque is written as N.m, or N \(\\times\) m, or N m.
A negative power is used for compound units which are formed by dividing one unit by another. Thus the unit of velocity is \(\\dfrac{\\text{metre}}{\\text{second}}\), expressed as \(\\text{m s}^{-1}\), and the unit of power is \(\\dfrac{\\text{joule}}{\\text{second}}\), expressed as \(\\text{J s}^{-1}\).
A unit in its short form is never written in plural. For example, \(10\) metres cannot be written as \(10\\ \\text{ms}\), because ms would mean millisecond.
To avoid powers of ten in the magnitude of a quantity, a prefix can be used with its unit. But a unit must not be written with more than one prefix. For example, instead of kMW we must write GW.
When a prefix is used with the symbol of a unit, the prefix and the symbol combined become the new symbol of the unit. For example, \(\\text{km}^{3}\) means \((10^{3}\\ \\text{m})^{3} = 10^{9}\\ \\text{m}^{3}\); it does not mean \(10^{3}\\ \\text{m}^{3}\).
That last rule is the one that quietly ruins calculations, because the cube applies to the prefix as well as to the unit. Table 1.9 puts the commonest slips beside their corrected forms.
1.8 Concept Of Least Count Of Common Instruments
In earlier classes you have already learned how to measure length, mass, time and temperature. For every such measurement of a physical quantity we rely on an instrument.
A metre rule, or a pair of vernier callipers, is used for length; a beam balance for mass; a watch for time; a thermometer for temperature; and an ammeter for electric current. Each instrument has a definite limit to the accuracy of the measurement it can give, and this limit is expressed in terms of its least count.
A measuring instrument is provided with a graduated scale, and the least count is the value of one smallest division on that scale. Reading the scale is therefore a matter of counting whole divisions, which is why nothing finer than one division can be read off with confidence.
Fig. 1.8-1 shows three familiar scales with their least counts worked out.
Diagram
Fig. 1.8-1 — Reading The Least Count Off Three Scales Three separate panels side by side, each showing part of a scale enlarged. Panel 1: a piece of a metre rule marked 0 cm to 3 cm with ten small divisions in each centimetre; caption arrow to one small division labelled ‘one division = 1 mm = 0.1 cm, so least count = 1 mm’. Panel 2: a stop watch dial with the marks 0 s and 5 s and ten equal divisions between them; arrow to one division labelled ‘one division = 0.5 s, so least count = 0.5 s’. Panel 3: an ammeter scale with the marks 0 A and 1 A and five equal divisions between them; arrow to one division labelled ‘one division = 0.2 A, so least count = 0.2 A’.
Take each in turn. The least count of a metre rule is the value of one of its divisions, which is one-tenth of a centimetre, or \(1\\ \\text{mm}\). The least count of a stop watch is \(0.5\\ \\text{s}\) if there are ten divisions between the \(0\) and \(5\\ \\text{s}\) marks. The least count of an ammeter having five divisions between the marks \(0\) and \(1\\ \\text{A}\) is \(0.2\\ \\text{A}\).
In every case the arithmetic is the same: take the value spanned by a set of divisions and divide it by the number of divisions in that set.
The smaller the least count of an instrument, the more precise is the measurement made by using it.
1.9 Measurement Of Length With A Metre Rule
Generally a metre rule, having its zero mark at one end and the \(100\\ \\text{cm}\) mark at the other end, is used to measure the length of an object. It has ten subdivisions in each one centimetre length, so the value of its one small division is \(1\\ \\text{mm}\), or \(0.1\\ \\text{cm}\).
Thus a metre rule can be used to measure length correct only up to \(1\\ \\text{mm}\), that is, up to one decimal place of a centimetre. It cannot measure length with still more accuracy, that is, up to the second decimal place of a centimetre.
The reason is shown in Fig. 1.9-1. If one end of the object lies between two small divisions on the metre rule, the mark nearer to the end is read, and so its length correct up to the second decimal point cannot be measured.
Diagram
Fig. 1.9-1 — Why A Metre Rule Stops At One Decimal Place An enlarged portion of a metre rule showing the marks 6 cm and 7 cm with the ten millimetre divisions between them. A rod is drawn along the rule with its left end exactly at the 6 cm mark and its right end lying between the 6.4 cm and 6.5 cm marks, closer to 6.4 cm. Draw a dotted vertical line from the rod’s end down to the scale and label the gap between that line and the 6.4 cm mark ‘this part cannot be measured — it is smaller than one division’. Caption arrows label ‘least count of metre rule = 1 mm = 0.1 cm’ and ‘reading recorded = 6.4 cm’.
However, it becomes possible to go further with the help of vernier callipers and a screw gauge. They are more accurate because they use two scales instead of one, and they have a least count smaller than \(0.1\\ \\text{cm}\). How a second scale achieves this is the subject of the next section.
1.10 Principle Of The Vernier
Pierre Vernier devised a method by which length up to the second decimal place of a centimetre — that is, correct up to \(0.1\\ \\text{mm}\) or \(0.01\\ \\text{cm}\) — can be measured. In this technique two scales are used. One scale, called the main scale, is fixed, while the other scale, called the vernier scale, slides along the main scale.
The main scale is graduated with the value of one division on it equal to \(1\\ \\text{mm}\). The graduations on the vernier scale are such that the length of \(n\) divisions on the vernier scale is equal to the length of \((n-1)\) divisions of the main scale.
Generally a vernier scale has ten divisions, and the total length of these ten divisions is equal to the length of \(10 - 1 = 9\) divisions of the main scale, that is, equal to \(9\\ \\text{mm}\). Thus each division of the vernier scale is of length \(0.9\\ \\text{mm}\) — smaller in size by \(\\frac{1}{10}\\ \\text{mm}\) than a division on the main scale. This difference is what is utilised as the least count for the measurement.
Diagram
Fig. 1.10-1 — Finding The Least Count Of A Vernier A main scale drawn horizontally, graduated in cm from 1 to 4 with ten small millimetre divisions in each centimetre, labelled ‘MAIN SCALE (FIXED)’ and marked ‘cm’ at the right end. Below it and touching it, a shorter vernier scale drawn with eleven marks numbered 0 to 10, its total length spanning exactly 9 divisions of the main scale, labelled ‘VERNIER SCALE (MOVABLE)’. Add a bracket above the main scale spanning one small division labelled ‘x = 1 mm’, and a bracket below the vernier spanning one vernier division labelled ‘0.9 mm’. Caption arrow: ‘the difference of 0.1 mm is the least count’.
1.10.1 Least Count Of A Vernier Or Vernier Constant
Since one vernier division is slightly shorter than one main scale division, the amount by which it falls short is the finest length the instrument can pick out.
This can be turned into a much more usable form. Let \(n\) divisions on the vernier be of length equal to that of \((n-1)\) divisions on the main scale, and let the value of one main scale division be \(x\). Then:
\[\\text{value of } n \\text{ divisions on vernier} = (n-1)x\]
so that, dividing both sides by \(n\),
\[\\text{value of 1 division on vernier} = \\frac{(n-1)x}{n}\]
Putting this into the definition above:
\[\\text{L.C.} = x - \\frac{(n-1)x}{n} = \\frac{nx - (n-1)x}{n} = \\frac{x}{n}\]
\[\\boxed{\\text{L.C.} = \\frac{\\text{value of one main scale division } (x)}{\\text{total number of divisions on vernier } (n)}}\]
where \(x\) is the value of one main scale division, usually in cm or mm, and \(n\) is the number of divisions on the vernier scale, a pure number. The result carries the same unit as \(x\). This form holds whenever the \(n\) vernier divisions are made equal in length to \((n-1)\) main scale divisions, which is how every vernier in your laboratory is built.
Thus the least count of a vernier is obtained simply by dividing the value of one division of the main scale by the total number of divisions on the vernier scale.
For the vernier shown in Fig. 1.10-1, the value of one division of the main scale is \(x = 1\\ \\text{mm}\) and the total number of divisions on the vernier is \(n = 10\). Hence:
\[\\text{L.C.} = \\frac{x}{n} = \\frac{1\\ \\text{mm}}{10} = 0.1\\ \\text{mm} = 0.01\\ \\text{cm}\]
1.10.2 Use Of A Vernier Scale
The two scales are so made that when the movable vernier scale touches the fixed end, its zero mark coincides with the zero mark of the main scale. Fig. 1.10-2 shows what happens when a rod is placed between them.
Diagram
Fig. 1.10-2 — Measurement With A Vernier Scale A horizontal main scale graduated in cm from 0 to 3 with millimetre subdivisions, labelled ‘MAIN SCALE (FIXED)’. At the left is a solid vertical block labelled ‘FIXED END’. A shaded rod, labelled ‘ROD’, lies along the scale with its left end against the fixed end. Touching the right end of the rod is the sliding vernier scale, labelled ‘VERNIER SCALE (MOVABLE)’, marked 0, 5 and 10. Show the vernier zero lying just beyond the 1.2 cm mark of the main scale; mark the small gap between the 1.2 cm mark and the vernier zero as ‘ab’. Show the 4th division of the vernier lining up exactly with a main scale mark, and draw a dotted vertical line through that coincidence.
The rod whose length is to be measured is placed along the main scale, and the vernier scale is moved so as to hold the rod between the fixed end and the movable vernier scale. In this position, the zero mark of the vernier scale is ahead of the \(1.2\\ \\text{cm}\) mark on the main scale.
Thus the actual length of the rod is \(1.2\\ \\text{cm}\) plus the extra length \(ab\), that is, the length between the \(1.2\\ \\text{cm}\) mark on the main scale and the zero mark on the vernier scale. This length \(ab\) cannot be measured by the main scale, since its value is less than one division marked on the main scale. This is exactly the difficulty the vernier was invented to solve.
To measure the length \(ab\), we first find the least count of the vernier scale. Here it is \(\\dfrac{0.1}{10}\\ \\text{cm} = 0.01\\ \\text{cm}\). We then note that \(p\)th division of the vernier scale which coincides, or is in line, with any division of the main scale. The product of this number of vernier divisions \(p\) with the least count gives the length \(ab\). This is called the vernier reading.
\[\\text{length } ab = \\text{vernier reading} = p \\times \\text{least count}\] \[\\text{total reading} = \\text{main scale reading} + \\text{vernier reading}\]
where \(p\) is the number of the coinciding vernier division, a pure number, and the least count is in cm, so that both readings and the total come out in cm.
Why The Coinciding Division Gives The Gap
The rule looks like a trick until you see where it comes from. If the \(p\)th division of the vernier coincides with some division of the main scale, then the length \(ab\) plus the length of \(p\) vernier divisions equals the length of \(p\) main scale divisions. Rearranging,
\[ab = p \\times (\\text{1 main scale division} - \\text{1 vernier scale division}) = p \\times \\text{L.C.}\]
So counting up to the coinciding line is really counting how many times the small difference between the two kinds of division has accumulated.
In Fig. 1.10-2, the main scale reading is \(1.2\\ \\text{cm}\) and the 4th division of the vernier scale coincides with a main scale division, so \(p = 4\) and the length \(ab\), that is the vernier reading, is \(4 \\times 0.01\\ \\text{cm} = 0.04\\ \\text{cm}\).
Hence the length of the rod is \(1.2\\ \\text{cm} + 0.04\\ \\text{cm} = 1.24\\ \\text{cm}\).
1.11 Vernier Callipers
A vernier callipers is also called slide callipers. It is used to measure the length of a rod, the diameter of a sphere, the internal and external diameters of a hollow cylinder, the depth of a small beaker or bottle, and similar small lengths.
1.11.1 Description Of The Instrument
A vernier callipers is shown in Fig. 1.11-1. It consists of a long and thin steel strip provided with a jaw \(J\_1\) at one end. On the strip a scale is graduated with the value of one division equal to \(1\\ \\text{mm}\); this is the main scale.
Diagram
Fig. 1.11-1 — The Parts Of A Vernier Callipers A labelled drawing of a vernier callipers lying horizontally. Label, in this order: the long steel strip carrying the MAIN SCALE, graduated in cm from 0 to 4; the FIXED JAW at the left of the main scale strip; the sliding strip carrying the VERNIER SCALE marked 0 to 10; the MOVABLE JAW on the sliding strip; the pair of lower jaws marked J₁ and J₂ together labelled OUTSIDE JAWS; the pair of upper jaws labelled INSIDE JAWS; the SCREW S on top of the sliding strip; and the thin strip T projecting from the right-hand end of the main scale strip, labelled STRIP (for depth). Show a small OBJECT held between the outside jaws.
Another small steel strip, provided with a jaw \(J\_2\) at its end, can slide over the main scale strip. This strip also has a scale graduated with ten divisions on it, the length of which is equal to \(9\\ \\text{mm}\). It is called the vernier scale.
For more precise measurement, the vernier scale can have \(20\), \(25\) or \(50\) divisions marked on it, and the total length of these vernier divisions will be equal to the length of one division less — that is, \(19\), \(24\) or \(49\) divisions respectively — on the main scale.
The vernier scale, which slides over the main scale, can be fixed at any position on the main scale with the help of a screw \(S\). Both jaws are parallel to each other and are projected on either side of the main scale to hold the given object.
The lower jaws are called the outside jaws, and they are used to measure the length of a rod, the diameter of a sphere or the external diameter of a cylinder. The upper jaws are called the inside jaws, which are used to measure the internal diameter of a hollow cylinder or pipe.
A vernier callipers is also provided with a thin and long strip \(T\) attached at the back of the vernier scale strip. It slides with the vernier scale at the back of the main scale strip. When the jaws \(J\_1\) and \(J\_2\) are in contact, the end of the strip \(T\) touches the end at the back of the main scale strip. The strip \(T\) is used to measure the depth of a small beaker or bottle.
The main parts of a vernier callipers, with their functions, are given in Table 1.10.
| Part | Function |
|---|---|
| Outside jaws | To measure the length of a rod, the diameter of a sphere, or the external diameter of a hollow cylinder. |
| Inside jaws | To measure the internal diameter of a hollow cylinder or pipe. |
| Strip | To measure the depth of a beaker or a bottle. |
| Main scale | To measure length correct up to \(1\\ \\text{mm}\). |
| Vernier scale | Helps to measure length correct up to \(0.1\\ \\text{mm}\). |
Table 1.10 — Vernier callipers: main parts and their functions.
1.11.2 Least Count Of Vernier Callipers
The least count of vernier callipers is equal to the difference between the values of one main scale division and one vernier scale division. It is calculated using the relation already obtained in section 1.10:
\[\\text{L.C.} = \\frac{\\text{value of one main scale division } (x)}{\\text{total number of divisions on vernier } (n)}\]
Two things therefore make the least count of a vernier callipers smaller, and so the instrument more precise: increasing the number of divisions on the vernier scale, and decreasing the value of one division on the main scale.
1.11.3 Zero Error In Vernier Callipers
On bringing the movable jaw \(J\_2\) in contact with the fixed jaw \(J\_1\), the zero mark of the vernier scale should coincide with the zero mark of the main scale. In this position the tenth division of the vernier scale coincides with the ninth division of the main scale. If it is so, the vernier is said to be free from zero error. In this condition the end of strip \(T\) also touches the end of the main scale strip.
But sometimes there is a mechanical error in a vernier callipers, due to which the zero mark of the vernier scale does not coincide with the zero mark of the main scale when the two jaws \(J\_1\) and \(J\_2\) are in contact. It is then said to have zero error.
The zero error is of two kinds: positive zero error, and negative zero error.
1.11.3.1 Positive Zero Error
On bringing the two jaws together, if the zero mark of the vernier scale is on the right of the zero mark of the main scale, the zero error is said to be positive. Fig. 1.11-2 shows the two scales of a vernier callipers with positive zero error. The positive zero error is equal to the distance of the zero mark of the vernier scale from the zero mark of the main scale.
Diagram
Fig. 1.11-2 — Positive Zero Error The two jaws J₁ and J₂ drawn closed at the left. Above them the MAIN SCALE graduated with 0 and 1 cm marked and millimetre divisions. Below it the VERNIER SCALE with divisions numbered 0 to 10, drawn so that its zero mark lies clearly to the right of the main scale zero. Draw a dotted vertical line through the 6th vernier division showing it in line with a main scale division, and label that line ‘COINCIDING DIVISION’. Caption label under the vernier: ‘positive zero error’.
To find this error, we note that division of the vernier scale which coincides with any division of the main scale. The number of this vernier division, when multiplied by the least count of the vernier, gives the zero error.
For the scales shown in Fig. 1.11-2, the least count is \(0.01\\ \\text{cm}\) and the 6th division of the vernier scale coincides with a main scale division. Therefore:
\[\\text{zero error} = +,6 \\times \\text{L.C.} = +,6 \\times 0.01\\ \\text{cm} = +,0.06\\ \\text{cm}\]
1.11.3.2 Negative Zero Error
On bringing the two jaws together, if the zero mark of the vernier scale is to the left of the zero mark of the main scale, the zero error is said to be negative. Fig. 1.11-3 shows the two scales of a vernier callipers with negative zero error. The negative zero error is equal to the length between the zero mark of the vernier scale and the zero mark of the main scale.
Diagram
Fig. 1.11-3 — Negative Zero Error The two jaws J₁ and J₂ drawn closed at the left. Above them the MAIN SCALE graduated with 0 and 1 cm marked and millimetre divisions. Below it the VERNIER SCALE with divisions numbered 0 to 10, drawn so that its zero mark lies clearly to the left of the main scale zero. Draw a dotted vertical line through the 6th vernier division showing it in line with a main scale division, and label that line ‘COINCIDING DIVISION’. Caption label under the vernier: ‘negative zero error’.
To find this error, we again note the division of the vernier scale which coincides with any division of the main scale. The number of this vernier division is subtracted from the total number of divisions on the vernier scale, and then the difference is multiplied by the least count.
In Fig. 1.11-3, the least count is \(0.01\\ \\text{cm}\), the sixth division of the vernier scale coincides with a certain division of the main scale, and the total number of divisions on the vernier is \(10\). Therefore:
\[\\text{zero error} = -,(10 - 6) \\times \\text{L.C.} = -,4 \\times 0.01\\ \\text{cm} = -,0.04\\ \\text{cm}\]
The two cases are set side by side in Table 1.11.
1.11.4 Correction Due To Zero Error
To get the correct reading, the zero error with its proper sign is always subtracted from the observed reading:
\[\\text{correct reading} = \\text{observed reading} - \\text{zero error (with sign)}\]
Since subtracting a positive quantity makes a reading smaller and subtracting a negative quantity makes it larger, this single rule covers both cases. Thus the positive zero error gets subtracted from the observed reading, while the negative zero error gets added to the observed reading.
1.11.5 Measuring The Length Of An Object With Vernier Callipers
Procedure
- Find the least count, and then the zero error of the vernier callipers if any.
- Move the jaw \(J\_2\) away from the jaw \(J\_1\) and place the object to be measured between the jaws \(J\_1\) and \(J\_2\). Move the jaw \(J\_2\) towards the jaw \(J\_1\) till it touches the object. Tighten the screw \(S\) to fix the vernier scale in its position.
- Note the main scale reading.
- Note that division \(p\) on the vernier scale which coincides, or is in line, with any division of the main scale. Multiply this vernier division \(p\) with the least count. This is the vernier scale reading, that is, vernier scale reading \(= p \\times \\text{L.C.}\)
- Add the vernier scale reading to the main scale reading. This gives the observed length.
- Repeat it three times and record the observations as below.
Observations
Total number of divisions on vernier scale, \(n = \\ldots\) Value of one division on main scale, \(x = \\ldots\\ \\text{cm}\) Least count, \(\\text{L.C.} = \\dfrac{x}{n} = \\ldots\\ \\text{cm}\) Zero error \(= \\ldots\\ \\text{cm}\)
| S. No. | Main scale reading \(a\) (in cm) | Vernier division coinciding, \(p\) | Vernier scale reading \(b = p \\times \\text{L.C.}\) (in cm) | Observed length \(= a + b\) (in cm) |
|---|---|---|---|---|
| 1. | ||||
| 2. | ||||
| 3. |
Table 1.12 — Observation table for measuring length with vernier callipers. Mean observed length \(= \\ldots\\ \\text{cm}\).
Taking three readings and averaging them matters, because a single reading can be spoilt by holding the object slightly askew between the jaws. From the mean observed length, subtract the zero error, if any, with its proper sign, to obtain the true measurement of the length of the given object. Thus:
\[\\text{observed length} = \\text{main scale reading} + (\\text{vernier division } p \\text{ coinciding with any division on the main scale}) \\times \\text{least count}\] \[\\text{true length} = \\text{observed length} - \\text{zero error (with sign)}\]
Fig. 1.11-4 illustrates how to read a vernier callipers in practice.
Diagram
Fig. 1.11-4 — Reading A Vernier Callipers A main scale graduated in cm from 4 to 7 with millimetre subdivisions, labelled ‘MAIN SCALE’. Below it the sliding vernier scale marked 0 to 10, labelled ‘VERNIER SCALE’, with an OBJECT shown held to its left. Place the vernier zero just past the 4.7 cm mark of the main scale. Draw a dotted vertical line through the 3rd vernier division showing it exactly in line with a main scale division.
1.12 Principle Of A Screw
An ordinary screw has threads on it at an equal distance along its length. On rotating the head of the screw, it moves forward or backward linearly along its axis. This is the whole trick of the instrument in the next section: a turning motion, which is easy to divide finely, is converted into a straight-line motion, which is what we want to measure.
1.12.1 Pitch Of A Screw
The linear distance which the screw moves in one complete round of its head is equal to the distance between two consecutive threads on it. This distance is called the pitch of the screw, and it is marked in Fig. 1.12-1.
Diagram
Fig. 1.12-1 — Pitch And Head Scale Of A Screw A screw drawn horizontally with clear threads along its shaft and a large cylindrical head at the right, labelled ‘HEAD’. Mark the distance between two consecutive thread crests with a small double-headed arrow labelled ‘PITCH’. To the right, draw the head seen end-on as a circle with equal divisions marked around its circumference, numbered 0, 25, 50, 75, and label it ‘circular or head scale, 100 equal divisions’. Add a curved arrow around the head labelled ‘one complete rotation’ and a straight arrow along the axis labelled ‘screw advances by one pitch’.
Generally the pitch of a screw is \(1\\ \\text{mm}\) or \(0.5\\ \\text{mm}\).
1.12.2 Least Count Of A Screw
The linear or axial movement of a screw can be used to measure small lengths. For this purpose the head of the screw is made large, and it is graduated along its circumference. Normally it has \(50\) or \(100\) equal divisions on it. This is called the circular or head scale.
Now follow what one division of that head scale is worth. If the pitch of a screw is \(1\\ \\text{mm}\) and it has \(100\) divisions on its head, then on rotation of \(100\) divisions of its circular scale — that is, one full turn — the pointed end of the screw moves by a distance equal to \(1\\ \\text{mm}\). Hence the distance moved by the screw along its axis on rotation of one division of the circular scale will be:
\[\\frac{1\\ \\text{mm}}{100} = 0.01\\ \\text{mm} = 0.001\\ \\text{cm}\]
This is the least distance which can be measured by the movement of the screw, and it is therefore called its least count. Fig. 1.12-2 shows what a single division of the head scale amounts to.
Diagram
Fig. 1.12-2 — One Division Of The Head Scale The head of the screw drawn end-on as a large circle with 100 divisions marked around it, with 0 at the top. Shade the small sector between the 0 mark and the first division and draw an arrow from 0 to that first division labelled ‘turn of one division’. Beside it, draw the screw in side view with a tiny double-headed arrow at its tip labelled ‘advance = pitch ÷ 100 = 0.01 mm’. Add a note ‘the large head turns through a big angle; the tip moves a tiny distance’.
The least count of a screw is obtained by dividing the pitch of the screw by the total number of divisions on its circular scale:
\[\\text{L.C.} = \\frac{\\text{pitch of screw}}{\\text{total number of divisions on circular scale}}\]
where the pitch is a length, usually in mm, and the number of divisions is a pure number, so that the least count comes out in the same unit as the pitch.
Three Worked Cases
- If a screw moves by \(1\\ \\text{mm}\) in one rotation and it has \(100\) divisions on its circular scale, then the pitch of the screw is \(1\\ \\text{mm}\) and the least count of the screw is \(\\dfrac{1\\ \\text{mm}}{100} = 0.01\\ \\text{mm} = 0.001\\ \\text{cm}\).
- If a screw moves by \(1\\ \\text{mm}\) in two rotations and its circular scale has \(50\) divisions, then the pitch of the screw is \(\\frac{1}{2}\\ \\text{mm}\), or \(0.5\\ \\text{mm}\), and the least count of the screw is \(\\dfrac{0.5\\ \\text{mm}}{50} = 0.01\\ \\text{mm} = 0.001\\ \\text{cm}\).
- If a screw moves by \(1\\ \\text{mm}\) in two rotations and its circular scale has \(500\) divisions, then the pitch of the screw is \(\\frac{1}{2}\\ \\text{mm}\), or \(0.5\\ \\text{mm}\), and the least count of the screw is equal to \(\\dfrac{0.5\\ \\text{mm}}{500} = 0.001\\ \\text{mm}\), which is \(10^{-6}\\ \\text{m}\), or \(1\\ \\mu\).
Notice from cases 1 and 2 that two screws with quite different pitches can end up with the same least count, because what matters is the pitch shared out over the head divisions, not the pitch alone.
1.13 Screw Gauge
A screw gauge works on the principle of a screw. It is used to measure the diameter of a wire or the thickness of a paper, usually up to an accuracy of the third decimal place of a centimetre, that is, correct up to \(0.001\\ \\text{cm}\). A finer version of it, the micrometer screw gauge, has a least count of \(1\\ \\mu\), or \(0.0001\\ \\text{cm}\).
1.13.1 Description Of The Instrument
A screw gauge is shown in Fig. 1.13-1. It has a U-shaped frame with a flat end \(A\) called the stud at one end, and a nut \(N\) with a cylindrical sleeve at the other end. Both the nut and the sleeve are threaded from inside.
Diagram
Fig. 1.13-1 — The Parts Of A Screw Gauge A labelled drawing of a screw gauge. Draw the U-FRAME open to the right. At the left arm of the U mark the flat end A labelled STUD. From the right arm project the cylindrical SLEEVE carrying the NUT N, with a horizontal BASE LINE graduated in mm along it, labelled MAIN SCALE. Through the sleeve draw the SCREW with its flat end B facing the stud, labelling the shaft SPINDLE. At the right end of the screw draw the HOLLOW CYLINDER (THIMBLE) with graduations round its rim, labelled CIRCULAR SCALE, and beyond it the RATCHET. Mark a small gap between A and B where the object is placed.
A screw with its one flat end \(B\) can be moved inside the nut \(N\) by rotating its head, which is in the form of a hollow cylinder, or thimble, provided at the other end of the screw.
A reference line, or base line, graduated in mm, is drawn on the cylindrical sleeve, parallel to the axis of the screw. This forms the main scale. The hollow cylinder, or thimble, is also graduated and it is divided into \(100\) equal parts. This is the circular scale, or head scale.
The thimble is attached with a ratchet by a spring, and the screw is always rotated by turning the ratchet. As soon as the flat end \(B\) of the screw comes in contact either with the stud \(A\) or with the object placed in between \(A\) and \(B\), any further rotation of the ratchet does not move the screw linearly to press \(B\) against \(A\) or against the object. Thus the ratchet helps in holding the given object gently between the stud \(A\) and the end \(B\) of the screw.
That gentleness is not a nicety. A wire squeezed hard between the jaws would flatten slightly, and the instrument would faithfully report the flattened thickness.
The main parts of a screw gauge, with their functions, are given in Table 1.13.
| Part | Function |
|---|---|
| Ratchet | To advance the screw by turning it till the object is gently held between the stud and the spindle of the screw. |
| Sleeve | To mark the main scale and the base line. |
| Thimble | To mark the circular scale. |
| Main scale | To read length correct up to \(1\\ \\text{mm}\). |
| Circular scale | Helps to read length correct up to \(0.01\\ \\text{mm}\). |
Table 1.13 — Screw gauge: main parts and their functions.
1.13.2 Pitch And Least Count Of A Screw Gauge
The pitch of a screw gauge is the linear distance moved by its screw on the main scale when the circular scale is given one complete rotation.
The least count of a screw gauge, however, is the linear distance moved by its screw along the main scale when the circular scale is rotated by one division on it. Thus, the least count of a screw gauge is:
\[\\text{L.C.} = \\frac{\\text{pitch of the screw gauge}}{\\text{total number of divisions on its circular scale}}\]
where the pitch is a length, usually in mm, and the number of divisions is a pure number.
Making The Least Count Smaller
The least count of a screw gauge can be decreased by decreasing the pitch and by increasing the total number of divisions on the circular scale. The micrometer screw gauge in Fig. 1.13-2 does both.
Diagram
Fig. 1.13-2 — A Micrometer Screw Gauge A micrometer screw gauge drawn with its sleeve carrying a MAIN SCALE graduated from 0 to 5 cm with 20 divisions in each centimetre, and the NUT and SCREW shown at the left. Draw the circular head end-on as a large dial marked with divisions numbered 0, 50, 100, 150, 200, 250, 300, 350, 400, 450 up to 500, labelled CIRCULAR SCALE. Caption arrows: ‘one main scale division = 1/20 cm = 0.5 mm’, ‘one rotation advances the screw by one main scale division, so pitch = 0.5 mm’, ‘500 divisions on the circular scale, so least count = 0.001 mm’.
In Fig. 1.13-2, a screw gauge is designed to have \(20\) divisions in \(1\\ \\text{cm}\) on the main scale and \(500\) divisions on its circular scale. Thus the value of one division on the main scale is \(\\frac{1}{20}\\ \\text{cm} = 0.05\\ \\text{cm}\), or \(0.5\\ \\text{mm}\). The screw moves one division along the main scale in one complete rotation of the circular scale, so its pitch is \(0.5\\ \\text{mm}\) and the least count is:
\[\\text{L.C.} = \\frac{0.5\\ \\text{mm}}{500} = 0.001\\ \\text{mm} = 10^{-6}\\ \\text{m}\]
It is therefore called the micrometer screw gauge. Such a screw gauge is used where a high degree of accuracy is required, for example in optical measurements.
1.13.3 Zero Error In A Screw Gauge
On bringing the flat end \(B\) of the screw in contact with the stud \(A\), if the zero mark of the circular scale coincides with the base line of the main scale, the screw gauge is said to be free from zero error.
But sometimes, due to mechanical error, on bringing the flat end \(B\) in contact with the stud \(A\), the zero mark of the circular scale is either below or above the base line of the main scale. The screw gauge is then said to have a zero error. It is of two kinds: positive zero error, and negative zero error.
1.13.3.1 Positive Zero Error
If, on bringing the flat end \(B\) of the screw in contact with the stud \(A\), the zero mark on the circular scale is below the base line of the main scale, the zero error is said to be positive. Fig. 1.13-3 shows the position of the two scales in this case.
Diagram
Fig. 1.13-3 — Positive Zero Error In A Screw Gauge An enlarged view of the sleeve and thimble edge. Draw the horizontal BASE LINE of the sleeve, labelled ‘Base line’. Draw the curved edge of the circular scale beside it, marked with divisions 0, 5, 10 and 95, arranged so that the zero mark of the circular scale lies clearly below the base line. Draw a short arrow to the 5th division showing it lying exactly on the base line, and caption the figure ‘zero error is positive’.
To find it, we note that division of the circular scale which coincides with the base line. This number, when multiplied by the least count of the screw gauge, gives the value of the positive zero error.
In Fig. 1.13-3 the 5th division of the circular scale coincides with the base line. If the least count of the screw gauge is \(0.001\\ \\text{cm}\), then:
\[\\text{zero error} = +,5 \\times 0.001\\ \\text{cm} = +,0.005\\ \\text{cm}\]
1.13.3.2 Negative Zero Error
If, on bringing the flat end \(B\) of the screw in contact with the stud \(A\), the zero mark on the circular scale is above the base line of the main scale, the zero error is said to be negative. Fig. 1.13-4 shows the position of the two scales in this case.
Diagram
Fig. 1.13-4 — Negative Zero Error In A Screw Gauge An enlarged view of the sleeve and thimble edge. Draw the horizontal BASE LINE of the sleeve, labelled ‘Base line’. Draw the curved edge of the circular scale beside it, marked with divisions 0, 95 and 90, arranged so that the zero mark of the circular scale lies clearly above the base line. Draw a short arrow to the 95th division showing it lying exactly on the base line, and caption the figure ‘zero error is negative’.
To find it, we note the division of the circular scale coinciding with the base line. This number is subtracted from the total number of divisions on the circular scale, and the difference is then multiplied by the least count of the screw gauge.
In Fig. 1.13-4 the 95th division of the circular scale coincides with the base line, and the total number of divisions on the circular scale is \(100\). If the least count of the screw gauge is \(0.001\\ \\text{cm}\), then:
\[\\text{zero error} = -,(100 - 95) \\times 0.001\\ \\text{cm} = -,0.005\\ \\text{cm}\]
1.13.4 Correction Due To Zero Error
To find the correct reading, the zero error with its sign is subtracted from the observed reading. Thus:
\[\\text{correct reading} = \\text{observed reading} - \\text{zero error (with sign)}\]
1.13.5 Measuring The Diameter Of A Wire With A Screw Gauge
Procedure
- Find the least count and the zero error, if any, of the screw gauge.
- Turn the ratchet anticlockwise so as to obtain a gap between the stud \(A\) and the flat end \(B\). Place the wire in the gap between the stud \(A\) and the flat end \(B\). Then turn the ratchet clockwise so as to hold the given wire gently between the stud \(A\) and the flat end \(B\) of the screw.
- Note the main scale reading.
- Note that division \(p\) of the circular scale which coincides with the base line of the main scale. This circular scale division \(p\), when multiplied by the least count, gives the circular scale reading, that is, circular scale reading \(= p \\times \\text{L.C.}\)
- Add the circular scale reading to the main scale reading to obtain the total reading, that is, the observed diameter of the wire.
- Repeat it by rotating the wire in its position by \(90^\\circ\), that is in a perpendicular direction. Take two more observations at different places of the wire and record them in the table below.
A wire is rarely a perfect cylinder, so turning it through \(90^\\circ\) and moving along its length is how a squashed or uneven patch is caught rather than trusted.
Observations
Pitch of the screw \(= \\ldots\\ \\text{cm}\) Total number of divisions on the circular scale \(= \\ldots\) Least count of screw gauge, \(\\text{L.C.} = \\dfrac{\\text{pitch}}{\\text{total number of divisions on circular scale}} = \\ldots\\ \\text{cm}\) Zero error \(= \\ldots\\ \\text{cm}\)
| S. No. | Main scale reading \(a\) (in cm) | Circular scale reading \(b = p \\times \\text{L.C.}\), where \(p\) is the division of the circular scale in line with the base line (in cm) | Observed diameter \(= a + b\) (in cm) |
|---|---|---|---|
| 1. (i) in one direction | |||
| 1. (ii) in \(\\perp\) direction | |||
| 2. (i) | |||
| 2. (ii) | |||
| 3. (i) | |||
| 3. (ii) |
Table 1.14 — Observation table for measuring the diameter of a wire. Mean observed diameter \(= \\ldots\\ \\text{cm}\).
From the observed mean diameter, subtract the zero error, if any, with its sign to get the true diameter. Thus:
\[\\text{observed diameter} = \\text{main scale reading} + (\\text{circular scale division } p \\text{ coinciding with the base line}) \\times \\text{least count}\] \[\\text{true diameter} = \\text{observed diameter} - \\text{zero error (with sign)}\]
Fig. 1.13-5 illustrates how to read a screw gauge.
Diagram
Fig. 1.13-5 — Reading A Screw Gauge An enlarged view of the sleeve and thimble. Along the sleeve draw the base line graduated in mm with 0 and the divisions up to 3 mm visible, labelled ‘mm’. The edge of the thimble is drawn just past the 3 mm mark. On the curved circular scale mark the divisions 40, 42 and 45, with the 42nd division lying exactly on the base line; draw a dotted line along the base line to that division.
1.13.6 Backlash Error
Sometimes, due to wear and tear of the threads of the screw, it is observed that on reversing the direction of rotation of the thimble, the tip of the screw does not start moving in the opposite direction at once, but remains stationary for a part of the rotation. This causes an error in the observation, which is called the backlash error.
To avoid the backlash error, while taking the measurements the screw should be rotated in one direction only. If it is required to change the direction of rotation of the screw, do not change the direction of rotation at once. Move the screw still further, stop there for a while, and then rotate it in the reverse direction.
Three Instruments Compared
From the discussion in these sections we find that a metre rule, vernier callipers and a screw gauge have decreasing least counts, which leads to an increase in accuracy. Table 1.15 sets them out together.
1.14 Measurement Of Time And The Simple Pendulum
In offices and homes we commonly use a pendulum clock to note time. Such a clock works because a pendulum repeats its swing in equal intervals, over and over, without being told to. Here we shall study the principle of a simple pendulum.
1.14.1 The Simple Pendulum
This is an ideal case, because we cannot have a heavy mass having the size of a point, and we cannot have a string which has no mass. In practice, as shown in Fig. 1.14-1, a heavy solid iron or brass ball is suspended by a light but strong thread from a rigid support. The ball is called the bob.
Diagram
Fig. 1.14-1 — A Simple Pendulum A rigid support drawn as a hatched horizontal bar at the top, with the point of suspension marked S. From S a straight thread hangs, labelled STRING, carrying a solid ball at its lower end labelled BOB. Draw the bob in three positions: at the centre marked O, labelled ‘mean or rest position’, and at the two extremes marked A on the left and B on the right, shown with dashed strings. Draw a curved dashed arc through A, O and B with arrows showing the to and fro motion. Mark the distance from S to the centre of the bob at O as l, labelled ‘effective length’. Mark the distance OA as the amplitude.
When the bob from its mean position \(O\) is pulled to one side and then released, the pendulum is set in motion and the bob moves alternately on either side of its mean position. The positions \(A\) and \(B\) are its extreme positions on the left and right respectively.
The pendulum used in a clock is not a simple pendulum, but a compound pendulum — that is, a body capable of oscillating about a horizontal axis passing through it.
1.14.3 Measuring The Time Period Of A Simple Pendulum
To measure the time period of a simple pendulum, the bob is slightly displaced from its rest position \(O\) and is then released. It begins to move to and fro about its mean position \(O\) in a vertical plane along with the string. The order of the steps is set out in FC 1.3.
Flowchart
FC 1.3 — Steps In Finding The Time Period Of A Pendulum Set up the pendulum of measured effective length l → displace the bob slightly and release it → start the stop watch and count 20 complete oscillations → note the total time t → calculate the time period as T = t ÷ 20 → repeat the whole set for a different length l
The time \(t\) for \(20\) complete oscillations is measured with the help of a stop watch, and dividing \(t\) by \(20\) gives the time period \(T\). The experiment is then repeated for different lengths of the pendulum, and the observations are recorded as in Table 1.16.
Why time twenty oscillations rather than one? Because the least count of a stop watch is either \(1\\ \\text{s}\) or \(0.5\\ \\text{s}\), so it cannot record a time period of \(1.2\\ \\text{s}\) or \(1.4\\ \\text{s}\) directly. Timing \(20\) oscillations and dividing by \(20\) makes such a fraction possible.
| \(S.\\ No.\) | Length \(l\) (in cm) | Time for 20 oscillations \(t\) (in s) | Time period \(T = \\dfrac{t}{20}\) (in s) | \(\\dfrac{l}{T^{2}}\) (in \(\\text{cm s}^{-2}\)) |
|---|---|---|---|---|
| 1 | 25 | 20 | 1.0 | 25 |
| 2 | 36 | 24 | 1.2 | 25 |
| 3 | 49 | 28 | 1.4 | 25 |
| 4 | 64 | 32 | 1.6 | 25 |
| 5 | 81 | 36 | 1.8 | 25 |
| 6 | 100 | 40 | 2.0 | 25 |
Table 1.16 — Time period for a pendulum of different lengths.
From observations 1 and 6 it can be noted that if the length of a pendulum is made four times, the period of oscillation gets doubled — that is, it now takes twice the time for one complete to and fro motion. Thus the time period \(T\) is directly proportional to the square root of the effective length \(l\) of the pendulum, \(T \\propto \\sqrt{l}\). Equivalently, the square of the time period \(T^{2}\) is directly proportional to the length \(l\), that is \(T^{2} \\propto l\), or \(\\dfrac{l}{T^{2}}\) is a constant. The last column of Table 1.16 shows exactly that constant.
1.14.4 Graph Of The Square Of Time Period Against Length
If a graph is plotted for the square of the time period \(T^{2}\) taken on the Y-axis against the length \(l\) taken on the X-axis, it comes out to be a straight line inclined to the \(l\)-axis, as shown in Fig. 1.14-2. This shows that \(T^{2}\) is directly proportional to \(l\).
Diagram
Fig. 1.14-2 — Graph Showing The Variation Of T² With l Draw the X-axis labelled ‘l (in cm)’ and the Y-axis labelled ‘T² (in s²)’, meeting at the origin. Draw a straight line rising from the origin to the upper right. Mark two points P and Q on the line, P above and to the right of Q. From P drop a normal to the X-axis meeting it at c and draw a normal to the Y-axis meeting it at a; from Q drop a normal to the X-axis meeting it at d and a normal to the Y-axis meeting it at b. Label the Y-axis values at a and b as T₁² and T₂², and the X-axis values at c and d as l₁ and l₂. Mark the point R where the horizontal through Q meets the vertical through P, so that PR is vertical and QR is horizontal.
1.14.5 Slope Of The T² Versus l Graph
The slope of the straight line obtained in the \(T^{2}\) versus \(l\) graph shown in Fig. 1.14-2 can be obtained by taking two points \(P\) and \(Q\) on the straight line and drawing normals from these points on the X and Y axes. Then note the values of \(T^{2}\), say \(T\_1^{2}\) and \(T\_2^{2}\) at \(a\) and \(b\) respectively, and also the values of \(l\), say \(l\_1\) and \(l\_2\) respectively at \(c\) and \(d\). Then:
\[\\text{slope} = \\frac{PR}{QR} = \\frac{ab}{cd} = \\frac{T\_1^{2} - T\_2^{2}}{l\_1 - l\_2}\]
This slope is found to be a constant at a place, and is equal to \(\\dfrac{4\\pi^{2}}{g}\), where \(g\) is the acceleration due to gravity at that place. Thus the value of \(g\) can be determined at a place from these measurements by using the following relation:
\[g = \\frac{4\\pi^{2}}{\\text{slope of } T^{2} \\text{ versus } l \\text{ graph}}\]
where the slope is in \(\\text{s}^{2},\\text{m}^{-1}\) if \(l\) is plotted in metre, so that \(g\) comes out in \(\\text{m s}^{-2}\).
1.14.6 Factors Affecting The Time Period Of A Simple Pendulum
From experiments on a simple pendulum, it is observed that:
- The time period of oscillation is directly proportional to the square root of its effective length, that is \(T \\propto \\sqrt{l}\). In other words, the square of the time period of oscillation, \(T^{2}\), is directly proportional to its effective length \(l\).
A pendulum clock has a compound pendulum made up of a metal such as brass or steel. Due to seasonal change of temperature, the effective length of the pendulum changes, and so the clock goes fast in winter and slow in summer. In winter, due to contraction, the effective length of the pendulum gets shortened, so its time period is decreased and the pendulum completes more oscillations in a given time — the clock goes fast. In summer, because of expansion, the effective length is increased, so its time period is increased and it completes fewer oscillations in a given time — the clock goes slow.
Similarly, while swinging, if we stand on the swing, the time period of the swing decreases, that is, the swing moves faster. This is because when we stand, the centre of gravity rises, so the effective length of the swing decreases, due to which its time period decreases.
- The time period of oscillation is inversely proportional to the square root of the acceleration due to gravity, that is \(T \\propto \\dfrac{1}{\\sqrt{g}}\).
For this reason a pendulum clock goes slow — that is, the time period of oscillation increases — when it is taken to mountains or into mines, because the value of \(g\) decreases both with altitude above the earth’s surface and with depth below it.
The time period of oscillation does not depend on the mass or material of the body suspended, that is, of the bob. If we take two pendulums of equal lengths, but with bobs of different masses or of different materials, their time periods will remain the same.
The time period of oscillation does not depend on the extent of swing on either side, that is, on the amplitude, provided the swing is not too large.
1.14.7 Expression For The Time Period Of A Simple Pendulum
The two dependences above are brought together in a single relation. The time period of oscillation of a simple pendulum is given by:
\[T = 2\\pi \\sqrt{\\frac{l}{g}} \\qquad \\text{or} \\qquad T^{2} = 4\\pi^{2},\\frac{l}{g}\]
where \(T\) is the time period in s; \(l\) is the effective length of the pendulum in m, equal to \((L + r)\), where \(L\) is the length of the string and \(r\) is the radius of the bob; and \(g\) is the acceleration due to gravity in \(\\text{m s}^{-2}\). The relation holds provided the swing is not too large.
Notice that neither the mass of the bob nor the amplitude appears anywhere in it, which is exactly what observations 3 and 4 said.
At a given place, since \(g\) is constant, the time period of a pendulum of given length is constant. This is why a pendulum can be used to measure time.
1.14.8 Seconds’ Pendulum
The pendulum of a clock which we use to note time in our house is a seconds’ pendulum. It takes time \(1\\ \\text{s}\) in moving from one extreme to the other extreme, so its time period is \(2\\ \\text{s}\).
The effective length of the seconds’ pendulum, at a place where \(g = 9.8\\ \\text{m s}^{-2}\), which is the average value, is nearly \(1\\ \\text{metre}\).
1.15 Summary
Measurement is the process of comparing a given physical quantity with a known standard quantity of the same nature. The standard chosen is called the unit.
A unit is a quantity of constant magnitude used to measure the magnitudes of other quantities of the same nature.
Every measurement has two parts, a numerical value and a unit: \(\\text{physical quantity} = (\\text{numerical value}) \\times (\\text{unit})\). Change the unit and the number changes, but the quantity does not.
A unit should be of convenient size, defined without ambiguity, reproducible, and unchanging with space and time. The last three conditions are essential for international acceptance.
A fundamental unit is independent of any other unit. A derived unit depends on, or can be expressed in terms of, the fundamental units — as the units of area, volume and speed are.
The older systems of unit are the C.G.S. (centimetre, gram, second), the F.P.S. (foot, pound, second) and the M.K.S. (metre, kilogram, second). All are now replaced by the S.I. system.
The S.I. system has seven fundamental units — metre (m) for length, kilogram (kg) for mass, second (s) for time, kelvin (K) for temperature, candela (cd) for luminous intensity, ampere (A) for electric current and mole (mol) for amount of substance — and two supplementary units, radian (rad) for angle and steradian (sr) for solid angle.
A prefix attached to a unit stands for a fixed power of ten, such as kilo for \(10^{3}\), mega for \(10^{6}\), milli for \(10^{-3}\) and micro for \(10^{-6}\).
The S.I. unit of length is the metre. Since 1983 one metre is the distance travelled by light in vacuum in \(\\dfrac{1}{299{,}792{,}458}\) of a second.
Sub units of the metre: \(1\\ \\text{cm} = 10^{-2}\\ \\text{m}\), \(1\\ \\text{mm} = 10^{-3}\\ \\text{m}\), \(1\\ \\mu = 10^{-6}\\ \\text{m}\) and \(1\\ \\text{nm} = 10^{-9}\\ \\text{m}\). Its multiple unit is the kilometre, \(1\\ \\text{km} = 10^{3}\\ \\text{m}\).
Non-metric units of length: \(1\\ \\text{A.U.} = 1.496 \\times 10^{11}\\ \\text{m}\), \(1\\ \\text{light year} = 9.46 \\times 10^{15}\\ \\text{m}\), \(1\\ \\text{parsec} = 3.08 \\times 10^{16}\\ \\text{m} = 3.26\\ \\text{light year}\), \(1\\ \\text{Å} = 10^{-10}\\ \\text{m}\) and \(1\\ \\text{fermi} = 10^{-15}\\ \\text{m}\).
A light year is a distance, not a time: it is the distance travelled by light in vacuum in one year.
The S.I. unit of mass is the kilogram. Its sub units are the gram, \(1\\ \\text{g} = 10^{-3}\\ \\text{kg}\), and the milligram, \(1\\ \\text{mg} = 10^{-6}\\ \\text{kg}\); its multiples are the quintal, \(100\\ \\text{kg}\), and the metric tonne, \(1000\\ \\text{kg}\).
One atomic mass unit is one-twelfth of the mass of one carbon-12 atom, and equals \(1.66 \\times 10^{-27}\\ \\text{kg}\). One solar mass, the mass of the sun, is \(2 \\times 10^{30}\\ \\text{kg}\).
The S.I. unit of time is the second. It was first defined as the \(\\dfrac{1}{86400}\)th part of a mean solar day, and since 1964 as the time of \(9{,}192{,}631{,}770\) vibrations of the radiation of the transition between two unperturbed ground state hyperfine levels of the cesium-133 atom.
Smaller units of time: \(1\\ \\text{ms} = 10^{-3}\\ \\text{s}\), \(1\\ \\mu\\text{s} = 10^{-6}\\ \\text{s}\), \(1\\ \\text{shake} = 10^{-8}\\ \\text{s}\) and \(1\\ \\text{ns} = 10^{-9}\\ \\text{s}\). Bigger units: \(1\\ \\text{day} = 86400\\ \\text{s}\) and \(1\\ \\text{year} = 3.1536 \\times 10^{7}\\ \\text{s}\).
A leap year has \(366\) days. Years divisible by \(4\) are leap years, except century years, which are leap years only if they are divisible by \(400\).
Derived units are obtained by replacing each quantity in a definition by its own unit, giving \(\\text{m s}^{-1}\) for speed, \(\\text{kg m s}^{-2}\) or newton for force, \(\\text{kg m}^{2},\\text{s}^{-2}\) or joule for work, \(\\text{kg m}^{-1},\\text{s}^{-2}\) or pascal for pressure, and so on.
The symbol of a unit named after a scientist takes a capital first letter (N, J, W, Pa, Hz, C, V), while the full name of every unit is written with a small first letter. A unit symbol is never written in the plural, and a unit carries only one prefix. When a prefix is used, prefix and symbol form a new symbol, so \(1\\ \\text{km}^{3} = 10^{9}\\ \\text{m}^{3}\).
The least count of an instrument is the smallest measurement that can be taken accurately with it, and is the value of one smallest division on its scale. The smaller the least count, the more precise the measurement.
A metre rule has a least count of \(1\\ \\text{mm}\), so it can measure length correct only up to one decimal place of a centimetre.
In a vernier, \(n\) divisions of the vernier scale are equal in length to \((n-1)\) divisions of the main scale. Its least count, also called the vernier constant, is the difference between one main scale division and one vernier scale division:
\[\\text{L.C.} = \\frac{x}{n}\]
where \(x\) is the value of one main scale division, usually in cm, and \(n\) is the number of divisions on the vernier scale.
- For a vernier callipers, with \(p\) the number of the coinciding vernier division:
\[\\text{observed length} = \\text{main scale reading} + p \\times \\text{L.C.}\]
A vernier callipers measures the length of a rod, the diameter of a sphere and external diameters with its outside jaws, internal diameters with its inside jaws, and the depth of a beaker with its strip.
Zero error is the length between the zero of the main scale and the zero of the vernier scale when the jaws are closed. It is positive if the vernier zero lies to the right of the main scale zero, and negative if it lies to the left. In every case:
\[\\text{correct reading} = \\text{observed reading} - \\text{zero error (with sign)}\]
- The pitch of a screw is the distance moved along its axis in one complete rotation of its head. The least count of a screw is the distance moved along the axis when the circular scale is rotated by one division:
\[\\text{L.C.} = \\frac{\\text{pitch}}{\\text{total number of divisions on the circular scale}}\]
A screw gauge works on the principle of a screw and measures the diameter of a wire or the thickness of a sheet, correct up to \(0.001\\ \\text{cm}\). Its ratchet holds the object gently; its zero error is positive when the circular scale zero is below the base line, and negative when above it.
Backlash error is caused by wear and tear of the threads, so that the tip does not move at once when the direction of rotation is reversed. It is avoided by rotating the screw in one direction only.
A metre rule (\(1\\ \\text{mm}\)), vernier callipers (\(0.1\\ \\text{mm}\)) and a screw gauge (\(0.01\\ \\text{mm}\)) have decreasing least counts, which leads to increasing accuracy.
A simple pendulum is a heavy point mass, the bob, suspended from a rigid support by a massless and inextensible string. Its effective length \(l\) is measured from the point of suspension to the centre of gravity of the bob.
One oscillation is one complete to and fro motion. The time period \(T\) is the time for one oscillation, in s, and the frequency \(f\) is the number of oscillations per second, in Hz:
\[f = \\frac{1}{T} \\qquad \\text{or} \\qquad T = \\frac{1}{f}\]
The amplitude is the maximum displacement of the bob from its mean position, measured in m.
Experiment shows that \(T \\propto \\sqrt{l}\), so \(T^{2} \\propto l\) and \(\\dfrac{l}{T^{2}}\) is constant at a place. A graph of \(T^{2}\) against \(l\) is a straight line through the origin, whose slope equals \(\\dfrac{4\\pi^{2}}{g}\), so that:
\[g = \\frac{4\\pi^{2}}{\\text{slope of } T^{2} \\text{ versus } l \\text{ graph}}\]
where \(g\) is the acceleration due to gravity in \(\\text{m s}^{-2}\).
- The time period of a simple pendulum is:
\[T = 2\\pi \\sqrt{\\frac{l}{g}}\]
where \(T\) is the time period in s, \(l\) the effective length in m and \(g\) the acceleration due to gravity in \(\\text{m s}^{-2}\). The relation holds provided the swing is not too large.
The time period does not depend on the mass or material of the bob, nor on the amplitude provided the swing is small. It increases with length and decreases as \(g\) increases, which is why a pendulum clock goes slow on mountains and in mines.
A seconds’ pendulum has a time period of \(2\\ \\text{s}\), taking \(1\\ \\text{s}\) to move from one extreme to the other. Its effective length is nearly \(1\\ \\text{m}\) where \(g = 9.8\\ \\text{m s}^{-2}\).
Flowchart
FC 1.4 — From Instrument To A Trustworthy Length Choose the instrument whose least count suits the length → calculate its least count from its scales → close the jaws and find the zero error with its sign → place the object and note the main scale reading → note the coinciding division and multiply it by the least count → add the two to get the observed reading → subtract the zero error with its sign to get the true length
1.16 Key Terms
- Amplitude — the maximum displacement of the bob of a pendulum from its mean position on either side; unit: metre (m).
- Angstrom (Å) — a unit of length equal to \(10^{-10}\\ \\text{m}\), used for atomic sizes and wavelengths of light.
- Astronomical unit (A.U.) — a unit of length equal to the mean distance between the earth and the sun, \(1.496 \\times 10^{11}\\ \\text{m}\).
- Atomic mass unit (a.m.u. or u) — one-twelfth of the mass of one carbon-12 atom, equal to \(1.66 \\times 10^{-27}\\ \\text{kg}\).
- Backlash error — the error caused when a worn screw fails to move at once on reversing the direction of rotation of its head.
- Centimetre (cm) — a sub unit of length equal to \(10^{-2}\\ \\text{m}\).
- Century — a unit of time of 100 years, equal to \(3.16 \\times 10^{9}\\ \\text{s}\).
- Day — the time taken by the earth to rotate once on its own axis, equal to \(86400\\ \\text{s}\).
- Decade — a unit of time of 10 years, equal to \(3.1536 \\times 10^{8}\\ \\text{s}\).
- Effective length of a pendulum — the distance of the centre of gravity of the bob from the point of suspension; unit: metre (m).
- Fermi (f) — a unit of length equal to \(10^{-15}\\ \\text{m}\), used for the size of a nucleus.
- Frequency of oscillation — the number of oscillations made in one second; unit: hertz (Hz), or \(\\text{s}^{-1}\).
- Gram (g) — a sub unit of mass equal to \(10^{-3}\\ \\text{kg}\).
- Hour (h) — a unit of time of 60 minutes, equal to \(3600\\ \\text{s}\).
- Kilometre (km) — a multiple unit of length equal to \(10^{3}\\ \\text{m}\).
- Leap year — a year in which February has 29 days, so that the year is of 366 days.
- Least count — the smallest measurement that can be taken accurately with an instrument; it carries the unit of the quantity being measured.
- Light year (ly) — the distance travelled by light in vacuum in one year, equal to \(9.46 \\times 10^{15}\\ \\text{m}\).
- Lunar month — the time of one lunar cycle, nearly 29.5 days.
- Main scale — the fixed graduated scale of a vernier callipers, or the base line scale on the sleeve of a screw gauge, read correct to \(1\\ \\text{mm}\).
- Mean solar day — the average time taken by the earth to complete one rotation on its own axis, equal to \(86400\\ \\text{s}\).
- Metric tonne — a multiple unit of mass equal to \(1000\\ \\text{kg}\), or 10 quintal.
- Micron (\(\\mu\)) — a sub unit of length equal to \(10^{-6}\\ \\text{m}\); also called the micrometre (\(\\mu\\text{m}\)).
- Millennium — a unit of time of 1000 years, equal to \(3.16 \\times 10^{10}\\ \\text{s}\).
- Milligram (mg) — a sub unit of mass equal to \(10^{-6}\\ \\text{kg}\), or \(10^{-3}\\ \\text{g}\).
- Millimetre (mm) — a sub unit of length equal to \(10^{-3}\\ \\text{m}\).
- Minute (min) — a unit of time equal to \(60\\ \\text{s}\).
- Month — a block of 28 to 31 days in the Gregorian calendar, taken to an approximation as 30 days, or \(2.592 \\times 10^{6}\\ \\text{s}\).
- Nanometre (nm) — a sub unit of length equal to \(10^{-9}\\ \\text{m}\).
- Numerical value — the number in a measurement, telling how many times the chosen unit is contained in the quantity; it is a pure number.
- Oscillation — one complete to and fro motion of the bob of a pendulum.
- Parsec — the distance from which the semi-major axis of the earth’s orbit subtends an angle of one second of arc, equal to \(3.08 \\times 10^{16}\\ \\text{m}\), or 3.26 light year.
- Physical quantity — anything about an object or an event that can be measured, such as length, mass or time.
- Pitch — the distance moved along its axis by a screw in one complete rotation of its head; unit: millimetre (mm).
- Prefix — a word placed before a unit, standing for a fixed power of ten, such as kilo for \(10^{3}\) or milli for \(10^{-3}\).
- Quintal — a multiple unit of mass equal to \(100\\ \\text{kg}\).
- S.I. system — the internationally agreed system of units, having seven fundamental units and two supplementary units.
- Screw gauge — an instrument working on the principle of a screw, used to measure the diameter of a wire or the thickness of a sheet, correct up to \(0.001\\ \\text{cm}\).
- Seconds’ pendulum — a pendulum whose time period of oscillation is \(2\\ \\text{s}\); its effective length is nearly \(1\\ \\text{m}\) where \(g = 9.8\\ \\text{m s}^{-2}\).
- Simple pendulum — a heavy point mass, the bob, suspended from a rigid support by a massless and inextensible string.
- Solar mass — the mass of the sun, equal to \(2 \\times 10^{30}\\ \\text{kg}\).
- Supplementary units — the radian (rad) for angle and the steradian (sr) for solid angle, added to the seven fundamental units of the S.I. system.
- System of units — a matched set of units chosen for the basic quantities, such as the C.G.S., F.P.S. and M.K.S. systems.
- Time period (period of oscillation) — the time taken to complete one oscillation; unit: second (s).
- Unit — a quantity of constant magnitude, used to measure the magnitudes of other quantities of the same nature.
- Vernier callipers — an instrument with a fixed main scale and a sliding vernier scale, used to measure length correct up to \(0.01\\ \\text{cm}\).
- Vernier scale — the sliding scale of a vernier, whose \(n\) divisions are equal in length to \((n-1)\) divisions of the main scale.
- Year (yr) — the time in which the earth completes one revolution around the sun, taken to an approximation as 365 days, or \(3.1536 \\times 10^{7}\\ \\text{s}\).
- Zero error — the length between the zero marks of the two scales of an instrument when its jaws, or its stud and screw end, are in contact; unit: centimetre (cm).
1.17 Points To Ponder
Every standard in this chapter is finally something that repeats itself or stays the same — a rotation, a wavelength, an atomic vibration. Which of the standards you have met depend on the earth itself, and what would happen to them if the earth’s rotation were slowly changing?
The definition of the metre changed in 1889, in 1960 and again in 1983. Does that mean every length recorded before those dates is now wrong? What is it about a redefinition that leaves earlier measurements standing?
The vernier reaches a finer reading not by engraving finer marks, but by making its divisions deliberately slightly out of step with the main scale. Why is spotting which two lines coincide easier for the eye than judging a fraction of a division? Can you think of another situation where a small mismatch is used to reveal something small?
A screw gauge reads to \(0.001\\ \\text{cm}\), but the wire it is measuring may not be perfectly round or uniform along its length. Is there any point in an instrument that is more precise than the object is consistent? How does the measuring procedure in section 1.13.5 respond to this problem?
A zero error can be removed exactly by arithmetic, but a backlash error cannot. What is the difference between an error that a calculation can undo and one that only a change of technique can avoid?
The time period of a pendulum does not depend on the mass of its bob. A heavier bob is harder to set moving, yet gravity also pulls it harder. Argue from those two facts why the mass might drop out of the answer altogether.
1.18 Objective Questions
- Which one of the following is a fundamental unit? (1 mark)
- newton
- candela
- pascal
- hertz
- One parsec is equal to: (1 mark)
- \(1.496 \\times 10^{11}\\ \\text{m}\)
- \(9.46 \\times 10^{15}\\ \\text{m}\)
- \(3.08 \\times 10^{16}\\ \\text{m}\)
- \(3 \\times 10^{8}\\ \\text{m}\)
- Which one of the following is a unit of time? (1 mark)
- fermi
- shake
- parsec
- candela
- The main scale of a vernier is graduated in millimetre and its vernier scale has \(20\) divisions. Its least count is: (1 mark)
- \(0.5\\ \\text{mm}\)
- \(0.05\\ \\text{mm}\)
- \(0.02\\ \\text{mm}\)
- \(0.005\\ \\text{mm}\)
- The pitch of a screw gauge is \(0.5\\ \\text{mm}\) and its circular scale has \(50\) divisions. Its least count is: (1 mark)
- \(1\\ \\text{mm}\)
- \(0.1\\ \\text{mm}\)
- \(0.01\\ \\text{mm}\)
- \(0.001\\ \\text{mm}\)
- Expressed in cubic metre, \(1\\ \\text{cm}^{3}\) is equal to: (1 mark)
- \(10^{-2}\\ \\text{m}^{3}\)
- \(10^{-3}\\ \\text{m}^{3}\)
- \(10^{-6}\\ \\text{m}^{3}\)
- \(10^{-9}\\ \\text{m}^{3}\)
- The time period of a simple pendulum depends on its: (1 mark)
- mass of bob
- material of bob
- effective length
- amplitude of swing
- If the effective length of a simple pendulum is made nine times, its time period becomes: (1 mark)
- nine times
- three times
- one-third
- unchanged
- Which one of the following is the smallest unit of length? (1 mark)
- nanometre
- angstrom
- micron
- fermi
Choose the answer based on the codes given below. a. Both A and R are true and R is the correct explanation of A b. Both A and R are true but R is not the correct explanation of A c. A is false but R is true d. A is true but R is false
Assertion (A): A light year is a unit of distance and not of time. Reason (R): It is obtained by multiplying the speed of light by a time interval of one year. (1 mark)
Assertion (A): For a screw gauge with a fixed number of divisions on its circular scale, reducing the pitch reduces the least count. Reason (R): The least count of a screw gauge is the pitch divided by the total number of divisions on the circular scale. (1 mark)
Assertion (A): A pendulum clock loses time when it is taken down into a deep mine. Reason (R): The value of the acceleration due to gravity increases with depth below the earth’s surface. (1 mark)
Match each quantity in column A with its unit in column B. (5 marks)
| Column A | Column B |
|---|---|
| a. Luminous intensity | i. steradian |
| b. Amount of substance | ii. hertz |
| c. Solid angle | iii. candela |
| d. Frequency | iv. pascal |
| e. Pressure | v. mole |
- Fill in the blanks.
- One astronomical unit is equal to ………… metre. (1 mark)
- The least count of a vernier is equal to the difference between one main scale division and one ………… division. (1 mark)
- A pendulum whose time period of oscillation is \(2\\ \\text{s}\) is called a ………… pendulum. (1 mark)
- The distance moved by a screw along its axis in one complete rotation of its head is called its ………… . (1 mark)
- State whether the following statements are true or false.
- The zero error of an instrument is always subtracted from the observed reading along with its own sign. (1 mark)
- The time period of a simple pendulum becomes larger when a heavier bob is used, the length being unchanged. (1 mark)
- A century year is a leap year only if it is divisible by 400. (1 mark)
- Name the following.
- The part of a vernier callipers used to measure the depth of a beaker. (1 mark)
- The error caused when the tip of a screw does not move at once on reversing the direction of rotation of the thimble. (1 mark)
- The instrument, out of a metre rule, vernier callipers and a screw gauge, that would be chosen to measure the thickness of a sheet of paper. (1 mark)
- Give one word or phrase for each of the following.
- The maximum displacement of the bob of a pendulum from its mean position. (1 mark)
- The smallest measurement that can be taken accurately with an instrument. (1 mark)
- State the S.I. unit of the following.
- Frequency of oscillation. (1 mark)
- Amount of substance. (1 mark)
- Effective length of a pendulum. (1 mark)
- Pick the odd one out and give a reason for your choice.
- Metre, kilogram, newton, second. (1 mark)
- Light year, parsec, astronomical unit, leap year. (1 mark)
1.19 Check Your Concepts
Define the term ‘unit’ and state the two parts in which the result of a measurement is expressed. (2 marks)
State the four properties that a unit chosen to measure a physical quantity should have. (2 marks)
Define the least count of an instrument and state its value for a metre rule. (2 marks)
Give a reason why a light year, although its name contains the word year, is not a unit of time. (2 marks)
State two ways in which the least count of a vernier callipers can be decreased. (2 marks)
Define the pitch of a screw gauge and state its relation with the least count. (2 marks)
Differentiate between a fundamental unit and a derived unit, stating the basis of difference, and give one example of each. (3 marks)
Differentiate between positive and negative zero error of a vernier callipers, stating the basis of difference, and state how each is accounted for. (3 marks)
Explain, with one example each, why the full name of a unit is written with a small first letter while the symbol of a unit named after a scientist begins with a capital letter. (3 marks)
Why is the time for 20 oscillations noted while finding the time period of a simple pendulum, instead of the time for one oscillation? (3 marks)
State the purpose of the ratchet in a screw gauge, and explain what would go wrong with the measured diameter of a thin wire if the screw were tightened without it. (3 marks)
What would happen to the time period of a simple pendulum if (i) its effective length were made four times, (ii) it were taken to a place where the acceleration due to gravity is smaller, and (iii) its brass bob were replaced by an iron bob of the same size? Give a reason in each case. (4 marks)
Explain why a metre rule cannot measure a length correct to the second decimal place of a centimetre, and describe how a vernier scale overcomes this limitation. (4 marks)
A pendulum clock keeps correct time in winter but loses time in summer. Explain this observation in terms of the factors on which the time period of a pendulum depends. (4 marks)
Describe, in steps, how you would measure the diameter of a wire using a screw gauge, stating the observations you would record and how the true diameter is finally obtained. (5 marks)
State the factors on which the time period of a simple pendulum depends and those on which it does not, write the relation for the time period naming each symbol with its unit, and state the condition under which the relation holds. (5 marks)
1.20 Diagrams, Graphs & Constructions
Draw a neat labelled diagram of a vernier callipers of the kind shown in Fig. 1.11-1, marking the outside jaws, the inside jaws, the main scale, the vernier scale, the screw and the strip. State the function of the inside jaws and of the strip. (5 marks)
Fig. 1.11-3 shows the two scales of a vernier callipers with the jaws closed. Redraw the scales, state which kind of zero error is shown, and explain how its value would be calculated if the least count were \(0.01\\ \\text{cm}\) and the vernier had \(10\) divisions. (4 marks)
Draw a neat labelled diagram of a screw gauge, of the kind shown in Fig. 1.13-1, marking the U-frame, the stud, the spindle, the sleeve with its base line, the thimble and the ratchet. (4 marks)
Fig. 1.13-5 shows the sleeve and thimble of a screw gauge in use. Redraw the figure for a gauge whose main scale reads \(2\\ \\text{mm}\) and whose 30th circular scale division lies on the base line, and state the reading in mm. (3 marks)
Draw a neat diagram of a simple pendulum of the kind shown in Fig. 1.14-1. Mark on it the point of suspension, the mean position, the two extreme positions, the effective length and the amplitude, and indicate one complete oscillation by arrows. (4 marks)
Sketch the graph you would expect if the square of the time period \(T^{2}\) of a simple pendulum were plotted against its effective length \(l\), as in Fig. 1.14-2. Label both axes with their units, state why the line passes through the origin, and explain how the acceleration due to gravity is obtained from its slope. (5 marks)
1.21 Practice With Numericals
The wavelength of a certain light is \(0.6\\ \\text{nm}\). Express it in (i) angstrom and (ii) metre. (2 marks)
A truck carries \(1.2\\ \\text{metric tonne}\) of cement. Express this mass in (i) kilogram and (ii) quintal. (2 marks)
A stop watch has \(20\) equal divisions between the \(0\) and \(10\\ \\text{s}\) marks. What is its least count? (2 marks)
The mass of one atom of an element is \(3.32 \\times 10^{-27}\\ \\text{kg}\). Express this mass in atomic mass unit, taking \(1\\ \\text{u} = 1.66 \\times 10^{-27}\\ \\text{kg}\). (2 marks)
The main scale of a vernier is graduated in millimetre and \(25\) divisions of its vernier scale are equal in length to \(24\) divisions of the main scale. Find its least count in mm and in cm. (3 marks)
A vernier callipers has a least count of \(0.01\\ \\text{cm}\) and is free from zero error. While measuring a rod, the main scale reading is \(5.2\\ \\text{cm}\) and the 7th vernier division coincides with a main scale division. Find the length of the rod. (3 marks)
A screw gauge has a pitch of \(1\\ \\text{mm}\) and \(100\) divisions on its circular scale, and is free from zero error. While measuring a wire, the main scale reads \(4\\ \\text{mm}\) and the 65th circular scale division lies on the base line. Find (i) the least count and (ii) the diameter of the wire in mm and in cm. (3 marks)
A simple pendulum completes \(50\) oscillations in \(100\\ \\text{s}\). Find (i) its time period, (ii) its frequency, and (iii) the name given to such a pendulum. (3 marks)
A vernier callipers has a least count of \(0.01\\ \\text{cm}\) and a zero error of \(+,0.04\\ \\text{cm}\). While measuring a cylinder, the main scale reading is \(3.1\\ \\text{cm}\) and the 9th vernier division coincides with a main scale division. Find (i) the observed diameter and (ii) the true diameter of the cylinder. (4 marks)
The screw of a screw gauge advances by \(1\\ \\text{mm}\) in two complete rotations, and its circular scale has \(50\) divisions. When the flat end of the screw is brought in contact with the stud, the zero of the circular scale lies above the base line and the 48th division is in line with it. Find (i) the pitch, (ii) the least count and (iii) the zero error of the screw gauge. (4 marks)
A screw has a pitch of \(0.5\\ \\text{mm}\). How many divisions should there be on its circular scale so that it can read correct up to \(0.002\\ \\text{mm}\)? (4 marks)
Find the time period of a simple pendulum of effective length \(0.49\\ \\text{m}\) at a place where \(g = 9.8\\ \\text{m s}^{-2}\). Take \(\\pi = 3.14\). (4 marks)
Two simple pendulums at the same place have effective lengths in the ratio \(9 : 1\). Find the ratio of their time periods, and state which of them makes more oscillations in one minute. (4 marks)
For a simple pendulum, the graph of \(T^{2}\) against \(l\) is a straight line of slope \(4.0\\ \\text{s}^{2},\\text{m}^{-1}\). Find the value of the acceleration due to gravity at that place, taking \(\\pi = 3.14\). (5 marks)