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*“When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is meagre and of an *unsatisfactory* kind.”* **– Lord Kelvin**

#### Measurements:

- Physics is a science of measurement. A measurement is a quantitative description of one or more fundamental properties compared to a standard.
- The measurement of a quantity is mentioned in two parts, the first part gives how many times of the standard unit and the second part gives the name of the unit. e.g. 5 m.
- The numerical value of a physical quantity is inversely proportional to its unit. For example: centimetre (cm) is a smaller unit compared to metre (m) and 5 m = 500 cm. We can see that the larger number is associated with the smaller unit and the smaller number is associated with the larger unit. If n
_{1}and n_{2}are the numerical values of a physical quantity in two different units say u_{1}and u2 respectively. then

n_{1} (u_{1}) = n_{2} (u_{2})

- The measure of a physical quantity depends on the system of units used.

#### Need for Measurement:

- Measurement is that operation by which we compare a physical quantity with a unit chosen for that quantity.
- 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.

#### Types of Measurements:

- Depending on the method, measurements are classified into two types.

a) Direct measurement and b) Indirect measurement

#### Direct Measurement:

- When measurements are taken directly using tools, instruments, or other calibrated measuring devices, they are called direct measurements.
- e.g. Measurement of length of a table by metre scale.

#### Indirect Measurement:

- 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.

#### Units of Measurements:

- For any measurement, a number and unit are required. When we say that the time is 5 second, then we mean that the said time is 5 times a certain standard time called 1 second.
- A unit is a selected magnitude of a physical variable in terms of which other magnitudes of the same variable can be expressed.
- Measurement without unit has no meaning.

#### Criteria for a Selection of Unit:

- The selection of unit depends on the magnitude of a quantity under consideration. For e.g. when we are measuring the diameter of a rod we should use millimetre as a unit. When we are measuring the height of a tower we should use the metre as a unit. When we are measuring the distance between the two cities we should use kilometre as a unit. When we are measuring the distance between the two stars we should use light years as a unit. This clearly indicates that when the magnitude of the measurement increases, then the unit used for the measurement should be larger.
- The unit should be neither too small nor too big in comparison with the physical quantity to be measured.
- The accuracy of measurement also influences the selection of unit. In the case of construction of a room where accuracy is not a major criteria metre or foot are used as units. But when constructing a rocket, accuracy is important hence millimetre or micrometre may be the unit. Thus when the accuracy is important then the unit used for the measurement should be smaller.

#### Requirements of Standard:

- The standard should be easily available.
- The standard should be non-destructible
- The standard should not change with the time
- The standard should not change with the place
- The standard should be easily reproducible

#### Characteristics of Standard Unit:

- It should be well defined without any doubt or ambiguity.
- It should be of suitable size. i.e. neither too long nor too small in comparison with quantity to be measured.
- It should be easily available.
- It should be non-destructible.
- It should not change with the time.
- It should not change with the place.
- It should be easily reproducible.

#### Old Methods of Measurements:

- To measure lengths units used were a finger, palm, span, cubit, foot, yard, fathom, furlong etc.

1 finger = 1 digit, 4 digits = 1 palm, 2 palms = 1 span, 2 spans = 1 cubit.

- 1 furlong: It was a length of a furrow an ox (Bullock) could plough without rest.

1 furlong = 220 yards

1 furlong = 0.201168 km

1 acre: It is an area an ox can plough in a day.

1 acre = 4840 sq. yards

1 acre = 40000 sq. feet

- We can see that, these standards may vary from person to person and animal to animal. Hence these units and standards are non-reliable.
- In 1799 after the revolution new republic of France accepted a metric system based on centimetre, gram and second (c.g.s. system). Britain accepted this system in 1852 for scientific purpose only.
- A committee consisting of chemist Antoine Laurent de Lavoisier and mathematician Joseph Louis Lagrange suggested the decimal system for measurement.
- In 1901 Italian engineer Giovanni Giorgi suggested a metric system based on metre, kilogram and second (M.K.S. system). It was upgraded to S.I. system by adding some more fundamental units in 1960.

#### Types of Physical Quantities:

- Physical quantities are those quantities which are measurable. The Physical quantities are classified as a) Fundamental quantities and b) Derived quantities

#### Fundamental Quantities:

- Fundamental quantities are those quantities which do not depend on other quantities for their measurements. The units of fundamental quantities are called as fundamental units.
- e.g. mass, length, time etc. are fundamental quantities. while, their units metre, kilogram, second etc. are fundamental units.
- Fundamental units can neither be derived from one another nor they can be further ressolved into other more simpler units.

#### Derived Quantities:

- Derived quantities are those quantities which depend on two or more other quantities for their measurements. The units of derived quantities are called as derived units
- e.g. density, acceleration, velocity, force, momentum, pressure etc. are derived quantities. while, their units kg m
^{-3}, m s^{-2}, m s^{-1}, newton, kg-m s^{-1}, pascal etc. are derived units.

### System of Units:

- 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 deine systems 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 of the system, the mass, the length and the time are considered to be fundamental quantities and their units are called as fundamental units.
- Following are some systems of units which are in common use.

**c.g.s. system of units**

Unit of length is centimetre (cm). Unit of mass is gram (g). Unit of time is second (s)

**m.k.s. system of units**

Unit of length is metre (m). Unit of mass is kilogram (kg). Unit of time is second (s)

**f.p.s. system of units**

Unit of length is a foot (ft). Unit of mass is a pound (Lb). Unit of time is second (s). This system is no more in use.

#### S.I. System of Units:

- In the year 1960, the Eleventh General Conference of Weights and Measures introduced International System of Units. The International Standard Organization (ISO) and 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.

- 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).

**Fundamental Units:**

Fundamental Quantity | S.I. Unit | Symbol | |

1 | Length | Metre | m |

2 | Mass | Kilogram | kg |

3 | Time | Second | s |

4 | Temperature | Kelvin | K |

5 | Electric current | Ampere | A |

6 | Luminous intensity | Candela | cd |

7 | Amount of substance | mple | mol |

**Supplementary Units**

Quantity | S.I. Unit | Symbol | |

1 | Plane angle | radian | rad |

2 | Solid angle | steradian | sr |

- This system of units is 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.

#### Guidelines For Writing 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).
- No full stop should be placed after the symbol.
- Index notation should be used to write derived unit. for example unit of velocity should be written as ms-1 instead of m/s.
- 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.
- Some space should be maintained between the number and its unit.

#### 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 examples 1 ampere = 1 volt / 1 ohm.
- It is a metric system of units. There is a decimal relationship between the units of 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.
- 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. 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.

#### 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 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

### 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.
**The reason of using Light as Standard:**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.

#### 1 kilogram:

- The unit of mass is a kilogram. Its symbol is ‘kg’.
- 1 kilogram is defined is total mass of 5.0188 × 10
^{25}atoms of C12 isotopes of carbon. Or The mass of a cylinder made up of platinum-iridium alloy kept at International Beuro of Weights and Measure is defined as 1 kilogram. - Reason for Using Platinum irridium 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 radiation corresponding to the transition between two hyperfine levels of the ground state of the Cesium-137 atom.
**Reason of Using Cesium – 137 as Standard:**Period of vibration of the atom of Cesium – 137 is used for defining the standard of time because the period of vibration of the atom of Cesium – 137 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 m
^{2}of a black body at the freezing point of platinum under pressure of 1.01325 × 10^{5}N/m^{2}.

#### 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 kilomole:

- The unit of amount of substance is kilomole. 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 10
^{23}. It is called as Avagadro’s number.

#### 1 radian:

- The unit of plane angle is radian
- 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.
- Its symbol is ‘rad’

#### 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.

### Other Important Units (Not Part of S.I. System):

- 1 fermi or femtometer (F) = 10
^{-15}m - 1 angstrom (A
^{0}) = 10^{-10}m - 1 micron or micrometre = 10
^{-6}m - 1 X-ray unit = 10
^{-13}m **Lightyear:**It is defined as a distance traveled by light in vacuum in one year. 1 light year = 9.46 x 10^{15}m**One astronomical unit:**It is a mean distance between the sun and the earth. 1 AU or 1 astronomical unit = 1.496 x 10^{11}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 10^{16}m- Note that light-year (ly) and parsec (pc) are units of distances and not of time.
**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- Extremely small areas are measured in barn. 1 barn = 10
^{-28}m^{2} - Chandra Shekhar limit is practical unit of measuring larg masses. 1 chandrashekhar unit = 1.4 times mass of the Sun.
- shake is used to measure very small time. 1 shake = 10
^{-8}s - Mass is measured in slug, metric ton, quintal. 1 slug = 14.57 kg, 1 metric ton = 1000 kg, 1 quintal = 100 kg.

### Units outside SI but Frequently Used in Physics:

Name | Symbol | Value in SI units |

minute (time) | min | 1 min = 60 s |

hour | h | 1 h = 60 min = 3600 s |

day | d | 1 d = 24 h = 86 400 s |

degree (angle) | ° | 1° = ( π/180) rad |

minute (angle) | ‘ | 1′ = (1/60)° = (π/10 800) rad |

second (angle) | ” | 1”= (1/60)’= (π/648 000) rad |

liter | L | 1 L = 1 dm^{3 }= 10^{-3} m^{3} |

metric ton | t | 1 t = 10^{3} kg |

neper | Np | 1 Np = 1 |

bel | B | 1 B = (1/2) ln 10 Np ^{(c)} |

unified atomic mass unit | u | 1 u = 1.660 54 x 10^{-27} kg, approximately |

astronomical unit | ua | 1 ua = 1.495 98 x 10^{11} m, approximately |

astronomical unit | ua | 1 ua = 1.495 98 x 10^{11} m, approximately |

**Units Currently Accepted for Use with the SI System:**

Name | Symbol | Value in SI units |

nautical mile | 1 nautical mile = 1852 m | |

knot | 1 nautical mile per hour = (1852/3600) m/s | |

are | a | 1 a = 1 dam^{2} = 10^{2} m^{2} |

hectare | ha | 1 ha = 1 hm^{2 }= 10^{4} m^{2} |

bar | bar | 1 bar = 0.1 MPa = 100 kPa = 1000 hPa = 10^{5} Pa |

ångström | Å | 1 Å = 0.1 nm = 10^{-10} m |

barn | b | 1 b = 100 fm^{2} = 10^{-28} m^{2} |

curie | Ci | 1 Ci = 3.7 x 10^{10} Bq |

roentgen | R | 1 R = 2.58 x 10^{-4} C/kg |

rad | rad | 1 rad = 1 cGy = 10^{-2} Gy |

electron volt | eV | 1 eV = 1.602 18 x 10^{-19} J, approximately |

## Some Derived Units

#### Prefixes Used in SI System:

### Examples to Understand Use of Units Numerical Problems

- Use of standard prefixes used in S.I. system to express following quantities
- 10
^{6}phones (1 Mphones) - 10
^{-6}( 1 μphones) - 10
^{12}( 1 Tcows) - 10
^{-9}monkeys( 1 nmonkeys) - 10
^{-12}birds ( 1 pbirds) - 12×10
^{-9}dogs ( 12 ndogs) - 34 x 10
^{3}boys (34 kboys)Numerical Problems

- 10
- Calculation of number of kilometers in 20 miles using the 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 × 5280 × 12 × 2.54 cm = 20 × 5280 × 12 × 2.54 ÷ 100 m =

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 × 10^{15} m

∴ 1 m = 1/(9.46 × 10^{15} ) light year

∴ 1 m = 1.057 × 10^{-16} light year

- Expression of 1 parsec in terms of light year Nume

1 parsec = 3.086 × 10^{16} m

∴ 1 parsec = 3.086 × 10^{16} × 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 10^{30}

- 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 10^{27}

- 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 10^{23}

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

Distance between the sun and the earth = 1.5 x 10^{11} m

Distance between the sun and the earth = (1.5 x 10^{11} m) ÷ (3.086 × 10^{16} ) = 4.861 × 10^{-6} parsec

Distance between the sun and the earth = 1.5 x 10^{11} × 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 unit of work

Work = Force × Displacement = mass × acceleration × displacement

∴ 1 J = kg × ms^{-2} × m

∴ 1 J = kg m^{2}s^{-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 + ct
^{2}, 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, unit of L.H.S. = Unit of R.H.S.

Hence unit of a, bt and ct^{2} is metre

Hence unit of a is metre

Unit of b × s = m

∴ Unit of b = m/s

Unit of c × s^{2} = m

Unit of c = m/s^{2}

**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.

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