Physics is a science of measurement. In science and engineering, we perform experiments. During experiments, we have to take readings. Thus all these experiments require some measurements to be made. During the production of mechanical products, we have to measure the parts so as to find whether the part is made as per the specifications. Thus measurements are necessary for production and quality control. A measurement is a quantitative description of one or more fundamental properties compared to a standard. To measure length is a very important step during the performance of experiments. Measurement can be done directly or indirectly. For direct methods metre scale, vernier callipers, micrometre screw gauges are used. In this article, we shall study the use of metre scale to measure length, diameter, etc.

When measurements are taken directly using tools, instruments, or other calibrated measuring devices, they are called direct measurements. e.g. Measurement of the length of a table by metre scale. When the measurement must be done through a formula or other calculations, the measurement is called indirect measurement. e.g. Measurement of the radius of the Earth.

Old Methods of Measurement of a Length

To measure lengths units used were a finger, palm, span, cubit, foot, yard, fathom, furlong etc.  The length of an inch was originally decided by Edward-II of England in the early 1300s as the length of three grains of barley laid end to end. Inch was divided into three parts called barley.

Measurement of Length:

The length is a fundamental quantity, it is used to measure a distance between two points in space. S.I. unit of length is a metre (m) and c.g.s. unit of length is centimetre (cm). Other practical units of measurement of length are micrometre (mm), millimetre (mm), kilometre (km), Angstrom (A°) etc.

Length of accessible objects can be measured directly using a metre scale, vernier callipers, micrometre screw gauge, measuring tapes, travelling microscopes etc. Length of non-accessible objects is measured indirectly.

Metre Scale:

one metre is divided into 100 equal parts, each part is called centimetre. Each centimetre is divided into 10 equal parts, each part is called millimetre. Engineering metre scale has centimetres marked on one edge and inches marked on another edge. Centimetres have decimal divisions while inches have fractional divisions. metre scale may have bevelled edges to avoid errors due to parallax.

Least count is the smallest measurement that can be taken accurately with an instrument or device. The least count of a metre scale is 1 mm or 0.1 cm.

Linear Measurements (Direct Method):

Place the metre scale along the object so that the ‘0’ mark on the metre scale coincides with one end of the object and reading at the other end of the scale indicates the length of the object. To avoid wear and tear off the end of the scale, sometimes the scale is placed along the object and readings at the ends of the object are taken. The length of the object is obtained by subtracting higher reading from lower reading.

The eye must be kept vertically above the end of the object so that the corresponding graduation can be read clearly. If the eye is not kept exactly vertically above the end of the object, it leads into error called parallax error.

In case, the end of the object lies between the two small divisions of the scale, the correct length is reported by noting the marking nearer to the end of the object. Limitation of metre scale is that it cannot measure the length of an object smaller than 1 mm or 0.1 cm.

Measurement of Length of a Curve

Thread Ruler Method (Direct Method):

In this case, a thread is laid along the curve. Then the length of the thread is measured using a metre scale.

Divider Ruler Method (Direct Method):

In this method, the curve is divided into small straight segments. Length of each such segment is measured using divider and scale. Then the total length of the curve can be obtained by adding lengths of all individual segments.

External diameter of cylinder or sphere

Blocks Ruler Method (Direct Method):

A cylinder or sphere whose external diameter is to be measured is placed between two blocks. The reading on the scale corresponding to the inner edge is x cm while that corresponding to the outer edge is y cm. Then the diameter is the absolute value of (x-y) cm.

Caliper Ruler Method (Direct Method):

To measure the external diameter of sphere or cylinder, an external calliper is used. In this method, the cylinder or sphere whose external diameter is to be measured is placed between two jaws of the calliper. The position of the calliper is fixed by tightening the screw. The distance between the jaws gives the external diameter of the cylinder or the sphere.

External Diameter of Cylinder: (Indirect Method):

A known number of turns (N) of a thin wire are wound on a cylinder whose diameter is to be measured. Then the wire is unwounded and straightened. Then its length is measured. The diameter of the rod can be obtained by the formula

To measure the internal diameter of the cylinder, an internal calliper is used.

External Diameter of Thin Wire: (Indirect Method):

A known number of turns of the wire whose diameter is to be measured are wound on a scale or on a rod of uniform diameter. The length of turns on the scale or the rod is measured. The diameter of the wire is calculated by dividing the length of turns by the number of turns.

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The quantity to be measured may be very large so that the SI unit of that quantity may be smaller to use or the quantity to be measured is so small that the SI unit of that quantity is very large to use. Hence other units were defined. Many of them are used with SI system of units. Examples: Light year is used to measure the distance between stars. unit fermi is used for measuring size of the nucleus of an atom. The radius of orbit of electrons in atom is measured in angstrom, etc.

• 1 fermi or femtometer (F) = 10^-15 m
• 1 angstrom (A⁰) = 10^-10 m
• 1 micron or micrometre = 10^-6 m
• 1 X-ray unit = 10^-13 m

Measuring Distances and Sizes:

• Lightyear: It is defined as a distance travelled by light in a vacuum in one year. 1 light year = 9.46 x 10¹⁵ m
• One astronomical unit: It is a mean distance between the sun and the earth. 1 AU or 1 astronomical unit = 1.496 x 10¹¹ m
• 1 parsec: It is the distance at which an arc of length one astronomical unit subtends an angle of 1 second of an arc. 1 parsec or parallactic second = 3.086 x 10¹⁶ m

Note that light-year (ly) and parsec (pc) are units of distances and not of time. A light-year (ly), parsec (pc) are used to measure the distance between stars and the astronomical unit is used to measure the distance between planets.

Measuring Masses:

• One a.m.u.: It is defined as 1/12 th of the mass of one atom. 1 a.m.u. = 1.66 x 10^-27 kg.
• Chandra Shekhar unit is practical unit of measuring large masses. 1 chandrashekhar unit = 1.4 times mass of the Sun.
• Mass is measured in slug, metric ton, quintal. 1 slug = 14.57 kg, 1 metric ton = 1000 kg, 1 quintal = 100 kg.

Measuring Small Areas:

• Extremely small areas are measured in barn. 1 barn = 10^-28 m²

Measuring Small Time:

• shake is used to measure very small time. 1 shake = 10^-8 s

Units outside SI but Frequently Used in Physics:

Units Currently Accepted for Use with the SI System:

Calculation of number of kilometers in 20 miles:

Conversion factors a mile = 5280 ft, 1 ft = 12 in, 1 in = 2.54 cm, 1 m = 100 cm, 1 km = 1000 m.

20 miles = 20 × 5280 ft = 20 × 5280 × 12 inches

20 miles = 20 × 5280 × 12 × 2.54 cm 

20 miles = 20 × 5280 × 12 × 2.54 ÷ 100  m

20 miles 20 × 5280 × 12 × 2.54 ÷ 100 ÷ 1000  km = 32.197 km

Calculation of the number of light years in one metre:

1 light year = distance travelled by light in one year

1 light year = 299, 792, 458 × 60 × 60 × 24 × 365 m

1 light year =9.46 × 10¹⁵ m

∴  1  m = 1/(9.46 × 10¹⁵ ) light year

∴  1  m = 1.057 × 10^-16  light year

Expression of  1 parsec in terms of light year:

1 parsec = 3.086 × 10¹⁶ m

∴ 1 parsec = 3.086 × 10¹⁶ × 1.057 × 10^-16  light year = 3.262 lightyears

The mass of an electron is 9.1 x 10^-31 kg. How many electrons would make 1 kg?

mass of electron = 9.1 x 10^-31 kg

∴ the number of electrons in 1 kg = 1/ (9.1 x 10^-31) = 1.099 x 10³⁰

The mass of an electron is 9.1 x 10^-31 kg. How many electrons would make 1 g?

mass of electron = 9.1 x 10^-31 kg = 9.1 x 10^-28 g

∴ the number of electrons in 1 g = 1/ (9.1 x 10^-28) = 1.099 x 10²⁷

The mass of a proton is 1.67 x 10^-27 kg. How many protons would make 1 g?

mass of proton = 1.67 x 10^-27  kg = 1.67 x 10^-24  g

∴ the number of protons in 1 g = 1/ (1.67 x 10^-24) = 5.99 x 10²³

Express the distance between the sun and earth in parsec and light year.

Distance between the sun and the earth = 1.5 x 10¹¹   m

Distance between the sun and the earth = (1.5 x 10¹¹ m)  ÷  (3.086 × 10¹⁶  ) = 4.861 × 10^-6  parsec

Distance between the sun and the earth = 1.5 x 10¹¹ × 1.057 × 10^-16  light year  = 1586 × 10^-5  light year

Derive the S.I. unit of joule (J) in terms of fundamental units.

Joule is the unit of work

Work = Force × Displacement = mass  × acceleration × displacement

∴   1  J = kg  × ms^-2 × m

∴   1  J = kg m²s^-2

A new unit of length is chosen such that the speed of light in a vacuum is unity. What is the distance between the sun and the earth in terms of a new unit if it takes 8 minutes and 20 seconds to cover the distance

Time taken by light = 8 min and 20 seconds = 8 × 60 + 20 = 480 + 20 = 500 seconds

Now 1 second of light corresponds to 1 new unit

Hence 500 seconds corresponds to 500 new units.

Hence the distance between the sun and the earth is 500 new units.

If x = a + bt + ct², where x is in metres and t is in seconds. Find units of a, b and c.

Physical quantities can only be added if they have the same unit.

Now, the unit of L.H.S. = The unit of R.H.S.

Hence unit of a, bt and ct² is metre

Hence unit of  a is metre

Unit of b ×  s  = m

∴   Unit of b  = m/s

Unit of c ×  s²  = m

Unit of c   = m/s²

Related Topics:

• 1.2.1 Introduction to Measurements
  • 1.2.1.1 Measurement and its Need
  • 1.2.1.2 System of Units
  • 1.2.1.4 Concept Application 01
• 1.2.2 Meassurement of Length, Area and Volume
• 1.2.3 Measurement of Mass, Weight, and Density
• 1.2.4 Measurement of Time
• 1.2.5 Dimensional Analysis
• 1.2.6 Error Analysis

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For More Topics in Physics Click Here

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There are as many units as there are independent quantities. We consider length, mass, and time three quantities which are independent of each other. Hence they have three separate units for their measurements. Hence it is required to define systems of units.

LIST OF SUB-TOPICS:

• 1.2.1.2.1 System of Units
• 1.2.1.2.2 SI System of Units
• 1.2.1.2.3 Writing SI Units
• 1.2.1.2. 4 Advantages of SI System
• 1.2.1.2.5 Finding Derived Units
• 1.2.1.2.6 Definitions of SI Units
• 1.2.1.2.7 Maintaining Uniformity of Standards
• 1.2.1.2.8 Prefixes Used in SI System
• 1.2.1.2.9 Responsibility of National Physical Laboratory

1.2.1.2.1 System of Units:

A system of units is a collection of units in which certain units are chosen as fundamental and all others are derived from them. This system is also called an absolute system of units. In most systems, the mass, length, and time are considered to be fundamental quantities, and their units are called fundamental units. The following are some systems of units which are in common use.

• c.g.s. system of units: The unit of length is centimetre (cm). The unit of mass is gram (g). The unit of time is second (s)
• m.k.s. system of units: The unit of length is the metre (m). The unit of mass is the kilogram (kg). The unit of time is second (s)
• f.p.s. system of units: The unit of length is a foot (ft). The unit of mass is a pound (Lb). The unit of time is second (s). This system is no more in use. This system is also known as Imperial system or the British Imperial system. Temperature is measured in Fahrenheit.

1.2.1.2.2 S.I. System of Units:

In the year 1960, the Eleventh General Conference of Weights and Measures introduced the International System of Units. The International Standard Organization (ISO) and the International Electrochemical Commission endorsed the system in 1962.  In October 1971 a replacement of the metric system of units was done with a new system called Systeme Internationale d’ Unites. The International System of Units, commonly known as the SI system, is the modern form of the metric system and is the most widely used system of measurement in the world. It provides a standard and coherent set of units for expressing physical quantities.

Fundamental Units:

Besides these seven basic units, there are two supplementary units. S.I. unit for the plane angle is radian (rad) and that of solid angle is steradian (sd).

Supplementary Units:

This system of units is an improvement and extension of the traditional metric system. Now, this system of units has replaced all other systems of units in all branches of science, engineering, industry, and technology.

1.2.1.2.3 Guidelines for Writing SI Units and Their Symbols:

• All units and their symbols should be written in small case letters e.g. centimetres (cm), metre (m), kilogram per metre cube ( kg m^-3).
• The units named after scientists are not written with a capital initial letter but its symbol is written in capital letter. Thus the unit of force is written as ‘newton’ or’ N’ and not as ‘Newton’. Similarly unit of work and energy is joule (J), S.I. unit of electric current is ampere (A). The S.I. nit of pressure is pascal (Pa) and that of temperature is kelvin (K).
• No full stop should be placed after the symbol.
• The denominators in a compound unit should be written with negative powers. Thus an index notation should be used to write a derived unit. for example unit of velocity should be written as ms^-1 instead of m/s. The unit of density is kilogram per metre cube ( kg m^-3 and not kg/m³)
• No plural form of a unit or its symbol should be used. example 5 newtons should be written as 5 N and not as 5 Ns.
• A compound unit obtained from units of two or more physical quantities is written either by putting a dot or leaving a space between symbols of two units. Example unit of torque is newton metre is written as Nm ot N.m. Unit of impulse is newton second is written as N s or N.s.
• Some space should be maintained between the number and its unit.

1.2.1.2.4 Advantages of S.I. System of Units:

• Units are simple to express
• This system uses only one unit for one physical quantity. Hence it is a rational system of units.
• Units of many physical quantities are related to each other through simple and elementary relationships   For example 1 ampere = 1 volt / 1 ohm.
• It is a metric system of units. There is a decimal relationship between the units of the same quantity and hence it is possible to express any small or large quantity as a power of 10. i.e. inter-conversion is very easy.    For e.g. 1kg   =  1000 gm  = 10³ gm
• The physical quantities can be expressed in terms of suitable prefixes.
• a joule is a unit of all forms of energy and it is a unit of work. Hence it forms a link between mechanical and electrical units. Hence S.I. the system is a rational system because it uses only one unit for one physical quantity.
• This system forms a logical and interconnected framework for all measurements in science, technology, and commerce.
• All derived units can be obtained by dividing and multiplying the basic and supplementary units and no numerical factors are introduced as in another system of units. Hence S.I. system of units is a coherent system. Hence S.I. system of units is used worldwide.

1.2.1.2.5 General Steps to Find Derived Unit:

• Step -1 Write the formula for the quantity whose unit is to be derived.
• Step -2 Substitute units of all the quantities in one system of units in their fundamental or standard form.
• Step -3 Simplify and obtain derive unit of the quantity.

Example: To find the unit of velocity.

Velocity is a derived quantity. Hence its unit is a derived unit.

The velocity is given by, velocity = displacement/time

S.I. unit of velocity = S.I. unit of displacement/ S.I. unit of time = m/s

Thus S.I. unit of velocity is m/s

1.2.1.2.6 Definitions of Fundamental Units in S. I. System:

1 metre:

• The unit of length is a metre.  Its symbol is ‘m’.
• The distance travelled by electromagnetic waves in the vacuum in 1/299, 792, 458 seconds is called 1 metre. The denominator is the velocity of light in the vacuum which is in m/s and is known accurately.
• One metre is 1,650, 763.73 times the wavelength of orange light emitte by a krypton atom at normal pressure. The wavelength of light is precisely defined in terms of electron transition in an atom, is easily reproducible and is not affected by the change in place, time, temperature and pressure, etc. Hence metre is defined in terms of wavelength of orange light.

1 kilogram:

• The unit of mass is a kilogram.  Its symbol is ‘kg’.
• 1 kilogram is defined is the total mass of 5.0188 × 1025 atoms of C12 isotopes of carbon. Or The mass of a cylinder made up of platinum-iridium alloy kept at the International Beuro of Weights and Measure is defined as 1 kilogram. Reason for Using Platinum iridium alloy for the cylinder is that it is least affected by environment and time.

1 second:

• The unit of time is second. Its symbol is ‘s’.
• 1 second is a time duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the Cesium-133 atom. Period of vibration of the atom of Cesium – 133  is used for defining the standard of time because the period of vibration of the atom of Cesium – 133 are precisely defined, is easily reproducible and is not affected by a change in place, time, temperature and pressure, etc.

1-degree kelvin:

• The unit of temperature is degree kelvin. Its symbol is ‘K’.
• 1-degree kelvin is a fraction 1/ 273.16 of the thermodynamic temperature of the triple point of the water. The triple point of the water is a temperature at which ice, water, and water vapour are in equilibrium.

1 candela:

• The unit of luminous intensity is candela. Its symbol is ‘cd’.
• 1 candela is luminous intensity in the normal direction of a surface of area 1/600000 m² of a black body at the freezing point of platinum under pressure of 1.01325 × 10⁵ N/m².

1 ampere:

• The unit of electric current is the ampere. Its symbol is ‘A’.
• 1 ampere is the constant current, which is maintained in each of two infinitely long straight parallel conductors of a negligible cross-section, situated one metre apart in vacuum, will produce between the conductors a force of 2 × 10^-7 N/m.

1 mole:

• The unit of the amount of substance is mole. Its symbol is ‘mol’
• 1 mole is the amount of substance which contains as many elementary entities (atoms, molecules, ions, electrons, etc.) as there are atoms in 0.012 kg of pure C12. The number of entities in one mole is 6.02252 X 10²³. It is called as Avagadro’s number.

1 radian:

• The unit of plane angle is the radian. Its symbol is ‘rad’
• One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.

1 steradian:

• The unit of solid angle is steradian. Its symbol is ‘sr’
• One steradian is defined as the solid angle that encloses a surface on the sphere of an area equal to the square of its radius.

1.2.1.2.7 Maintaining Uniformity of Standards

An international body Conference Generale des Poids et Measures or CGPM (General Conference of Weight and Measures) has been given the authority to decide the standards and units by international agreement. It holds its meetings and any change in the standard units are communicated through the publications of the Conference.

India adopted the metric system of units in 1956 by Parliament Act “Weights and Measures Act- 1956”. The function of manufacturing, maintaining, monitoring, and improving the standards of measurements is discharged by the National Physical Laboratory (NPL), New Delhi. The uniformity in standards is maintained as follows:

• Measures (e.g. balances and weights) used by shopkeepers are expected to be certified by the Department of Measures and Weights of the local government.
• The working standards of these local departments have to be calibrated against the state-level standards, or any laboratory which is entitled to do so.
• The state-level laboratories are required to get their standards calibrated from the National Physical Laboratory at the national level, which is equivalent to international standards. Thus, measurements made at any place in the world are connected with the international system.

1.2.1.2.8 Prefixes Used in SI System:

Examples to Understand the Use of Units in Numerical Problems

Use of standard prefixes used in S.I. system to express the following quantities:

• 10⁶ phones  (1 Mphones)
• 10^-6 phones ( 1 μphones)
• 10¹² cows ( 1 Tcows)
• 10^-9 monkeys( 1 nmonkeys)
• 10^-12 birds ( 1 pbirds)
• 12×10^-9 dogs ( 12 ndogs)
• 34 x 10³ boys (34 kboys)

1.2.1.2.9 Responsibility of National Physical Laboratory (NPL):

• It is the responsibility of the NPL to calibrate the measurement standards in these laboratories at different levels.
• The weights and balances used in local markets and other areas are expected to be certified by the Department of Weights and Measures of the local government.
• To strengthen and advance physics-based research and development for the overall development of science and technology in the country.
• To establish, maintain and improve continuously by research, for the benefit of the nation,
• To identify and conduct after due consideration, research in areas of physics which are most appropriate to the needs of the nation and for the advancement of the field
• To assist industries, national and other agencies in their developmental tasks by precision measurements, calibration, development of devices, processes, and other allied problems related to physics.
• To keep itself informed of and study critically the status of physics.

Related Topics:

• 1.2.1 Introduction to Measurements
  • 1.2.1.1 Measurement and its Need
  • 1.2.1.3 Other Important Units
  • 1.2.1.4 Concept Application 01
• 1.2.2 Meassurement of Length, Area and Volume
• 1.2.3 Measurement of Mass, Weight, and Density
• 1.2.4 Measurement of Time
• 1.2.5 Dimensional Analysis
• 1.2.6 Error Analysis

For More Topics in Introduction to Measurements Click Here

For More Topics in Physics Click Here

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