Contributed by:

The highlights are:

1. Physical Quantity – Fundamental & Derived Quantities

2. Unit – Fundamental & Derived Units

3. Characteristics of Standard Unit

4. FPS, CGS, MKS & SI System of Units

5. Definition of Fundamental SI units

6. Measurement of Length – Large Distances and Small Distances

7. Measurement of Mass and Measurement of Time

8. Accuracy, Precision of Instruments, and Errors in Measurements

9. Systematic Errors and Random Errors

10. Absolute Error, Relative Error, and Percentage Error

11. Combination of Errors, in Addition, Subtraction, Multiplication, Division, and Exponent.

12. Significant Figures, Scientific Notation, and Rounding off Uncertain Digits

13. Dimensions, Dimensional Formulae and Dimensional Equations

14. Dimensional Analysis – Applications and Demerits

1. Physical Quantity – Fundamental & Derived Quantities

2. Unit – Fundamental & Derived Units

3. Characteristics of Standard Unit

4. FPS, CGS, MKS & SI System of Units

5. Definition of Fundamental SI units

6. Measurement of Length – Large Distances and Small Distances

7. Measurement of Mass and Measurement of Time

8. Accuracy, Precision of Instruments, and Errors in Measurements

9. Systematic Errors and Random Errors

10. Absolute Error, Relative Error, and Percentage Error

11. Combination of Errors, in Addition, Subtraction, Multiplication, Division, and Exponent.

12. Significant Figures, Scientific Notation, and Rounding off Uncertain Digits

13. Dimensions, Dimensional Formulae and Dimensional Equations

14. Dimensional Analysis – Applications and Demerits

1.
UNITS AND MEASUREMENT

1. Physical Quantity – Fundamental & Derived Quantities

2. Unit – Fundamental & Derived Units

3. Characteristics of Standard Unit

4. fps, cgs, mks & SI System of Units

5. Definition of Fundamental SI units

6. Measurement of Length – Large Distances and Small Distances

7. Measurement of Mass and Measurement of Time

8. Accuracy, Precision of Instruments and Errors in Measurements

9. Systematic Errors and Random Errors

10. Absolute Error, Relative Error and Percentage Error

11. Combination of Errors in Addition, Subtraction, Multiplication, Division and

Exponent.

12. Significant Figures, Scientific Notation and Rounding off Uncertain Digits

13. Dimensions, Dimensional Formulae and Dimensional Equations

14. Dimensional Analysis – Applications- I, II & III and Demerits

Next

1. Physical Quantity – Fundamental & Derived Quantities

2. Unit – Fundamental & Derived Units

3. Characteristics of Standard Unit

4. fps, cgs, mks & SI System of Units

5. Definition of Fundamental SI units

6. Measurement of Length – Large Distances and Small Distances

7. Measurement of Mass and Measurement of Time

8. Accuracy, Precision of Instruments and Errors in Measurements

9. Systematic Errors and Random Errors

10. Absolute Error, Relative Error and Percentage Error

11. Combination of Errors in Addition, Subtraction, Multiplication, Division and

Exponent.

12. Significant Figures, Scientific Notation and Rounding off Uncertain Digits

13. Dimensions, Dimensional Formulae and Dimensional Equations

14. Dimensional Analysis – Applications- I, II & III and Demerits

Next

2.
Physical Quantity

A quantity which is measurable is called ‘physical quantity’.

Fundamental Quantity

A physical quantity which is the base and can not be derived from any

other quantity is called ‘fundamental quantity’.

Examples: Length, Mass, Time, etc.

Derived Quantity

A physical quantity which can be derived or expressed from base or

fundamental quantity / quantities is called ‘derived quantity’.

Examples: Speed, velocity, acceleration, force, momentum,

torque, energy, pressure, density, thermal conductivity,

resistance, magnetic moment, etc.

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A quantity which is measurable is called ‘physical quantity’.

Fundamental Quantity

A physical quantity which is the base and can not be derived from any

other quantity is called ‘fundamental quantity’.

Examples: Length, Mass, Time, etc.

Derived Quantity

A physical quantity which can be derived or expressed from base or

fundamental quantity / quantities is called ‘derived quantity’.

Examples: Speed, velocity, acceleration, force, momentum,

torque, energy, pressure, density, thermal conductivity,

resistance, magnetic moment, etc.

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3.
Measurement of any physical quantity involves comparison with a

certain basic, arbitrarily chosen, internationally accepted reference

standard called unit.

Fundamental Units

The units of the fundamental or base quantities are called fundamental

or base units.

Examples: metre, kilogramme, second, etc.

Derived Units

The units of the derived quantities which can be expressed from the

base or fundamental quantities are called derived units.

Examples: metre/sec, kg/m3, kg m/s2, kg m2/s2, etc.

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certain basic, arbitrarily chosen, internationally accepted reference

standard called unit.

Fundamental Units

The units of the fundamental or base quantities are called fundamental

or base units.

Examples: metre, kilogramme, second, etc.

Derived Units

The units of the derived quantities which can be expressed from the

base or fundamental quantities are called derived units.

Examples: metre/sec, kg/m3, kg m/s2, kg m2/s2, etc.

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4.
System of Units

A complete set of both fundamental and derived units is known as the

system of units.

Characteristics of Standard Units

A unit selected for measuring a physical quantity must fulfill the following

i) It should be well defined.

ii) It should be of suitable size i.e. it should neither be too large nor too

small in comparison to the quantity to be measured.

iii) It should be reproducible at all places.

iv) It should not change from place to place or time to time.

v) It should not change with the physical conditions such as temperature,

pressure, etc.

vi) It should be easily accessible.

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A complete set of both fundamental and derived units is known as the

system of units.

Characteristics of Standard Units

A unit selected for measuring a physical quantity must fulfill the following

i) It should be well defined.

ii) It should be of suitable size i.e. it should neither be too large nor too

small in comparison to the quantity to be measured.

iii) It should be reproducible at all places.

iv) It should not change from place to place or time to time.

v) It should not change with the physical conditions such as temperature,

pressure, etc.

vi) It should be easily accessible.

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5.
Various System of Units

In earlier time, various systems like ‘fps’, ‘cgs’ and ‘mks’ system of units were

used for measurement. They were named so from the fundamental units in

their respective systems as given below:

Quantity Dimension System of units

fps cgs mks

Length L foot centi metre metre

Mass M pound gramme kilogramme

Time T second second second

Systeme Internationale d’ unites (SI Units)

The SI system with standard scheme of symbols, units and abbreviations was

developed and recommended by General Conference on Weights and

Measures in 1971 for international usage in scientific, technical, industrial and

commercial work.

This is the system of units which is at present accepted internationally.

SI system uses decimal system and therefore conversions within the system

are quite simple and convenient. Home Next Previous

In earlier time, various systems like ‘fps’, ‘cgs’ and ‘mks’ system of units were

used for measurement. They were named so from the fundamental units in

their respective systems as given below:

Quantity Dimension System of units

fps cgs mks

Length L foot centi metre metre

Mass M pound gramme kilogramme

Time T second second second

Systeme Internationale d’ unites (SI Units)

The SI system with standard scheme of symbols, units and abbreviations was

developed and recommended by General Conference on Weights and

Measures in 1971 for international usage in scientific, technical, industrial and

commercial work.

This is the system of units which is at present accepted internationally.

SI system uses decimal system and therefore conversions within the system

are quite simple and convenient. Home Next Previous

6.
Fundamental Units in SI system

Quantity Symbol SI unit Symbol

Length L metre m

Mass M kilogramme kg

Time T second s

Electric Current A ampere A

Main units Thermodynamic K kelvin K

Temperature

Light Intensity Cd candela cd

Amount of mole mole mol

substance

Plane angle dθ radian rad

ary units Solid angle dΩ steradian sr

Home Next Previous

Quantity Symbol SI unit Symbol

Length L metre m

Mass M kilogramme kg

Time T second s

Electric Current A ampere A

Main units Thermodynamic K kelvin K

Temperature

Light Intensity Cd candela cd

Amount of mole mole mol

substance

Plane angle dθ radian rad

ary units Solid angle dΩ steradian sr

Home Next Previous

7.
Definition of SI Units

The metre is the length of the path travelled by light in vacuum during a

time interval of 1/299,792,458 of a second. (1983)

The kilogram is equal to the mass of the international prototype of the

kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of

Weights and Measures, at Sevres, near Paris, France. (1889)

The second is the duration of 9,192,631,770 periods of the radiation

corresponding to the transition between the two hyperfine levels of the

ground state of the cesium-133 atom. (1967)

The ampere is that constant current which, if maintained in two straight

parallel conductors of infinite length, of negligible circular cross-section,

and placed 1 metre apart in vacuum, would produce between these

conductors a force equal to 2×10–7 newton per metre of length. (1948)

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The metre is the length of the path travelled by light in vacuum during a

time interval of 1/299,792,458 of a second. (1983)

The kilogram is equal to the mass of the international prototype of the

kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of

Weights and Measures, at Sevres, near Paris, France. (1889)

The second is the duration of 9,192,631,770 periods of the radiation

corresponding to the transition between the two hyperfine levels of the

ground state of the cesium-133 atom. (1967)

The ampere is that constant current which, if maintained in two straight

parallel conductors of infinite length, of negligible circular cross-section,

and placed 1 metre apart in vacuum, would produce between these

conductors a force equal to 2×10–7 newton per metre of length. (1948)

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8.
The kelvin, is the fraction 1/273.16 of the thermodynamic

temperature of the triple point of water. (1967)

The candela is the luminous intensity, in a given direction, of a

source that emits monochromatic radiation of frequency 540×1012

hertz and that has a radiant intensity in that direction of 1/683 watt

per steradian. (1979)

The mole is the amount of substance of a system, which contains as

many elementary entities as there are atoms in 0.012 kilogramme of

carbon-12. (1971)

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temperature of the triple point of water. (1967)

The candela is the luminous intensity, in a given direction, of a

source that emits monochromatic radiation of frequency 540×1012

hertz and that has a radiant intensity in that direction of 1/683 watt

per steradian. (1979)

The mole is the amount of substance of a system, which contains as

many elementary entities as there are atoms in 0.012 kilogramme of

carbon-12. (1971)

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9.
Plane angle

Plane angle ‘dθ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’.

ds

r

ds

dθ dθ =

r r

O

Solid angle

Solid angle ‘dΩ’ is the ratio of the intercepted area ‘dA’ of the spherical

surface described at the apex ‘O’ as the centre, to the square of its radius ‘r’.

Its SI unit is ‘steradian’.

dA

r

r

dΩ dA

dΩ =

r2

O

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Plane angle ‘dθ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’.

ds

r

ds

dθ dθ =

r r

O

Solid angle

Solid angle ‘dΩ’ is the ratio of the intercepted area ‘dA’ of the spherical

surface described at the apex ‘O’ as the centre, to the square of its radius ‘r’.

Its SI unit is ‘steradian’.

dA

r

r

dΩ dA

dΩ =

r2

O

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10.
The following conventions are adopted while writing a unit:

(1) Even if a unit is named after a person the unit is not written in

letters. i.e. we write joules not Joules.

(2) For a unit named after a person the symbol is a capital letter e.g. for

joules we write ‘J’ and the rest of them are in lowercase letters e.g.

second is written as ‘s’.

(3) The symbols of units do not have plural form i.e. 70 m not 70 ms or

10 N not 10 Ns.

(4) Punctuation marks are not written after the unit

e.g. 1 litre = 1000 cc not 1000 c.c.

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(1) Even if a unit is named after a person the unit is not written in

letters. i.e. we write joules not Joules.

(2) For a unit named after a person the symbol is a capital letter e.g. for

joules we write ‘J’ and the rest of them are in lowercase letters e.g.

second is written as ‘s’.

(3) The symbols of units do not have plural form i.e. 70 m not 70 ms or

10 N not 10 Ns.

(4) Punctuation marks are not written after the unit

e.g. 1 litre = 1000 cc not 1000 c.c.

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11.
Some Units are retained for general use

(Though outside SI)

Name Symbol Value in SI Unit

minute min 60 s

hour h 60 min = 3600 s

day d 24 h = 86400 s

year y 365.25 d = 3.156 x 107 s

degree 0

10 = (π / 180) rad

litre l 1 dm3 = 10-3 m3

tonne t 103 kg

carat c 200 mg

bar bar 0.1 MPa = 105 Pa

curie ci 3.7 x 1010 s-1

roentgen r 2.58 x 10 -4 C/kg

quintal q 100 kg

barn b 100 fm2 = 10-28 m2

are a 1 dam2 = 102 m2

hectare ha 1 hm2 = 104 m2

standard atmosphere atm 101325 Pa = 1.013 x 105 Pa

pressure Home Next Previous

(Though outside SI)

Name Symbol Value in SI Unit

minute min 60 s

hour h 60 min = 3600 s

day d 24 h = 86400 s

year y 365.25 d = 3.156 x 107 s

degree 0

10 = (π / 180) rad

litre l 1 dm3 = 10-3 m3

tonne t 103 kg

carat c 200 mg

bar bar 0.1 MPa = 105 Pa

curie ci 3.7 x 1010 s-1

roentgen r 2.58 x 10 -4 C/kg

quintal q 100 kg

barn b 100 fm2 = 10-28 m2

are a 1 dam2 = 102 m2

hectare ha 1 hm2 = 104 m2

standard atmosphere atm 101325 Pa = 1.013 x 105 Pa

pressure Home Next Previous

12.
MEASUREMENT OF LENGTH

The order of distances varies from 10-14 m (radius of nucleus) to 1025 m

(radius of the Universe)

The distances ranging from 10-5 m to 102 m can be measured by direct

methods which involves comparison of the distance or length to be

measured with the chosen standard length.

i) A metre rod can be used to measure distance as small as 10-3 m.

ii) A vernier callipers can be used to measure as small as 10-4 m.

iii) A screw gauge is used to measure as small as 10-5 m.

For very small distances or very large distances indirect methods are used.

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The order of distances varies from 10-14 m (radius of nucleus) to 1025 m

(radius of the Universe)

The distances ranging from 10-5 m to 102 m can be measured by direct

methods which involves comparison of the distance or length to be

measured with the chosen standard length.

i) A metre rod can be used to measure distance as small as 10-3 m.

ii) A vernier callipers can be used to measure as small as 10-4 m.

iii) A screw gauge is used to measure as small as 10-5 m.

For very small distances or very large distances indirect methods are used.

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13.
Measurement of Large Distances P

The following indirect methods may be used

to measure very large distances: θ

1.Parallax method

D D

• Let us consider a far away planet ‘P’ at a

distance ‘D’ from our two eyes.

• Suppose that the lines joining the planet to

the left eye (L) and the right eye (R) subtend

an angle θ (in radians).

L b R

• The angle θ is called ‘parallax angle’ or

‘parallactic angle’ and the distance LR = b is

called ‘basis’.

LR b

θ= =

• As the planet is far away, b/D << 1, and D D

therefore θ is very small.

b

D=

•Then, taking the distance LR = b as a θ

circular arc of radius D, we have

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The following indirect methods may be used

to measure very large distances: θ

1.Parallax method

D D

• Let us consider a far away planet ‘P’ at a

distance ‘D’ from our two eyes.

• Suppose that the lines joining the planet to

the left eye (L) and the right eye (R) subtend

an angle θ (in radians).

L b R

• The angle θ is called ‘parallax angle’ or

‘parallactic angle’ and the distance LR = b is

called ‘basis’.

LR b

θ= =

• As the planet is far away, b/D << 1, and D D

therefore θ is very small.

b

D=

•Then, taking the distance LR = b as a θ

circular arc of radius D, we have

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14.
Measurement of the size or angular diameter of an astronomical

If ‘d’ is the diameter of the planet and ‘α’ is

d

the angular size of the planet (the angle

subtended by d at the Earth E), then

α = d/D

The angle α can be measured from the same

D D

location on the earth.

α

It is the angle between the two directions

when two diametrically opposite points of the

planet are viewed through the telescope.

E

Since D is known, the diameter d of the planet

can be determined from

d=αD

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If ‘d’ is the diameter of the planet and ‘α’ is

d

the angular size of the planet (the angle

subtended by d at the Earth E), then

α = d/D

The angle α can be measured from the same

D D

location on the earth.

α

It is the angle between the two directions

when two diametrically opposite points of the

planet are viewed through the telescope.

E

Since D is known, the diameter d of the planet

can be determined from

d=αD

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15.
Echo method or Reflection method

υ×t =2S

S= • This method is used to measure the

υ ×t

2

distance of a hill.

• A gun is fired towards the hill and the

time interval between the instant of firing

the gun and the instant of hearing the echo

of the gun shot is noted.

• This is the time taken by the sound to S

travel from the observer to the hill and S

back to the observer. Sound

wave

• If v = velocity of sound;

Gun fire Echo received

S = the distance of hill from the observer

and

T = total time taken, then

vxT

S=

2

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υ×t =2S

S= • This method is used to measure the

υ ×t

2

distance of a hill.

• A gun is fired towards the hill and the

time interval between the instant of firing

the gun and the instant of hearing the echo

of the gun shot is noted.

• This is the time taken by the sound to S

travel from the observer to the hill and S

back to the observer. Sound

wave

• If v = velocity of sound;

Gun fire Echo received

S = the distance of hill from the observer

and

T = total time taken, then

vxT

S=

2

Home Next Previous

16.
• In place of sound waves, LASER can be used to

measure the distance of the Moon from the Earth.

LASER is a monochromatic, intense and unidirectional

• If ‘t’ is the time taken for the LASER beam in going

to and returning from the Moon, then the distance can

be calculated from the formula

cxt

S= where c = 3 x 108 m s-1

2

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measure the distance of the Moon from the Earth.

LASER is a monochromatic, intense and unidirectional

• If ‘t’ is the time taken for the LASER beam in going

to and returning from the Moon, then the distance can

be calculated from the formula

cxt

S= where c = 3 x 108 m s-1

2

Home Next Previous

17.
Estimation of Very Small Distances

1. Using Electron Microscope:

For visible light the range of wavelengths is from about 4000 Å to 7000 Å

(1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve

particles with sizes smaller than this.

Electron beams can be focused by properly designed electric and magnetic

The resolution of such an electron microscope is limited finally by the fact

that electrons can also behave as waves.

The wavelength of an electron can be as small as a fraction of an angstrom.

Such electron microscopes with a resolution of 0.6 Å have been built. They

can almost resolve atoms and molecules in a material.

In recent times, tunneling microscopy has been developed in which again

the limit of resolution is better than an angstrom. It is possible to estimate

the sizes of molecules.

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1. Using Electron Microscope:

For visible light the range of wavelengths is from about 4000 Å to 7000 Å

(1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve

particles with sizes smaller than this.

Electron beams can be focused by properly designed electric and magnetic

The resolution of such an electron microscope is limited finally by the fact

that electrons can also behave as waves.

The wavelength of an electron can be as small as a fraction of an angstrom.

Such electron microscopes with a resolution of 0.6 Å have been built. They

can almost resolve atoms and molecules in a material.

In recent times, tunneling microscopy has been developed in which again

the limit of resolution is better than an angstrom. It is possible to estimate

the sizes of molecules.

Home Next Previous

18.
2. Avogadro’s Method:

A simple method for estimating the molecular size of oleic acid

is given below.

Oleic acid is a soapy liquid with large molecular size of the

order of 10–9 m.

The idea is to first form mono-molecular layer of oleic acid on

water surface.

We dissolve 1 cm3 of oleic acid in alcohol to make a solution of

20 cm3 (ml). Then we take 1 cm3 of this solution and dilute it to

20 cm3, using alcohol.

1

So, the concentration of the solution is cm3 of oleic

20 x 20

acid per cm3 of solution.

Next we lightly sprinkle some lycopodium powder on the

surface of water in a large trough and we put one drop of this

solution in the water.

The oleic acid drop spreads into a thin, large and roughly

circular film of molecular thickness on water surface.

Home Next Previous

A simple method for estimating the molecular size of oleic acid

is given below.

Oleic acid is a soapy liquid with large molecular size of the

order of 10–9 m.

The idea is to first form mono-molecular layer of oleic acid on

water surface.

We dissolve 1 cm3 of oleic acid in alcohol to make a solution of

20 cm3 (ml). Then we take 1 cm3 of this solution and dilute it to

20 cm3, using alcohol.

1

So, the concentration of the solution is cm3 of oleic

20 x 20

acid per cm3 of solution.

Next we lightly sprinkle some lycopodium powder on the

surface of water in a large trough and we put one drop of this

solution in the water.

The oleic acid drop spreads into a thin, large and roughly

circular film of molecular thickness on water surface.

Home Next Previous

19.
Then, we quickly measure the diameter of the thin film to get its

area A.

Suppose we have dropped ‘n’ drops in the water.

Initially, we determine the approximate volume of each drop

(V cm3).

Volume of n drops of solution = nV cm3

1

Amount of oleic acid in this solution = nV c

20 x 20 m3

This solution of oleic acid spreads very fast on the surface of

water and forms a very thin layer of thickness ‘t’.

If this spreads to form a film of area ‘A’ cm2, then the thickness

of the film

Volume of the film nV

t= or t= cm

Area of the film 20 x 20 x A

If we assume that the film has mono-molecular thickness, then

this becomes the size or diameter of a molecule of oleic acid. The

value of this thickness comes out to be of the order of 10 –9m.

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area A.

Suppose we have dropped ‘n’ drops in the water.

Initially, we determine the approximate volume of each drop

(V cm3).

Volume of n drops of solution = nV cm3

1

Amount of oleic acid in this solution = nV c

20 x 20 m3

This solution of oleic acid spreads very fast on the surface of

water and forms a very thin layer of thickness ‘t’.

If this spreads to form a film of area ‘A’ cm2, then the thickness

of the film

Volume of the film nV

t= or t= cm

Area of the film 20 x 20 x A

If we assume that the film has mono-molecular thickness, then

this becomes the size or diameter of a molecule of oleic acid. The

value of this thickness comes out to be of the order of 10 –9m.

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20.
Range of Lengths

The size of the objects we come across in the Universe varies over a very

wide range.

These may vary from the size of the order of 10–14 m of the tiny nucleus of an

atom to the size of the order of 1026 m of the extent of the observable

Universe.

We also use certain special length units for short and large lengths which are

given below:

Unit Symbol Value Definition

1 fermi 1f 10–15 m

1 angstrom 1Å 10–10 m

1 Astronomical 1 AU 1.496 × 1011 m Average distance of the Sun from the Earth

Unit

1 light year 1 ly 9.46 × 1015 m The distance that light travels with speed of

3 × 108 m s–1 in 1 year

1 parsec 3.08 × 1016 m The distance at which average radius of

(3.26 ly) Earth’s orbit subtends an angle of 1 arc

second

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The size of the objects we come across in the Universe varies over a very

wide range.

These may vary from the size of the order of 10–14 m of the tiny nucleus of an

atom to the size of the order of 1026 m of the extent of the observable

Universe.

We also use certain special length units for short and large lengths which are

given below:

Unit Symbol Value Definition

1 fermi 1f 10–15 m

1 angstrom 1Å 10–10 m

1 Astronomical 1 AU 1.496 × 1011 m Average distance of the Sun from the Earth

Unit

1 light year 1 ly 9.46 × 1015 m The distance that light travels with speed of

3 × 108 m s–1 in 1 year

1 parsec 3.08 × 1016 m The distance at which average radius of

(3.26 ly) Earth’s orbit subtends an angle of 1 arc

second

Home Next Previous

21.
Range and Order of Lengths (Llongest : Lshortest = 1041 : 1)

S.No Size of the object or distance Length (m)

1 Size of proton 10-15

2 Size of atomic nucleus 10-14

3 Size of the Hydrogen atom 10-10

4 Length of a typical virus 10-8

5 Wavelength of a light 10-7

6 Size of the red blood corpuscle 10-5

7 Thickness of a paper 10-4

8 Height of the Mount Everest from sea level 104

9 Radius of the Earth 107

10 Distance of the moon from the earth 108

11 Distance of the Sun from the earth 1011

12 Distance of the Pluto from the Sun 1013

13 Size of our Galaxy 1021

14 Distance of the Andromeda galaxy 1022

15 Distance of the boundary of observable universe 1026

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S.No Size of the object or distance Length (m)

1 Size of proton 10-15

2 Size of atomic nucleus 10-14

3 Size of the Hydrogen atom 10-10

4 Length of a typical virus 10-8

5 Wavelength of a light 10-7

6 Size of the red blood corpuscle 10-5

7 Thickness of a paper 10-4

8 Height of the Mount Everest from sea level 104

9 Radius of the Earth 107

10 Distance of the moon from the earth 108

11 Distance of the Sun from the earth 1011

12 Distance of the Pluto from the Sun 1013

13 Size of our Galaxy 1021

14 Distance of the Andromeda galaxy 1022

15 Distance of the boundary of observable universe 1026

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22.
MEASUREMENT OF MASS

The SI unit of mass is kilogram (kg).

The prototypes of the International standard kilogramme supplied by the

International Bureau of Weights and Measures (BIPM) are available in many

other laboratories of different countries.

In India, this is available at the National Physical Laboratory (NPL), New

While dealing with atoms and molecules, the kilogramme is an inconvenient

unit. In this case, there is an important standard unit of mass, called the

unified atomic mass unit (u), which has been established for expressing the

mass of atoms as

1 unified atomic mass unit = 1 u

One unified mass unit is equal to (1/12) of the mass of an atom of Carbon-12

isotope (12C6 ) including the mass of electrons.

1 u = 1.66 × 10–27 kg

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The SI unit of mass is kilogram (kg).

The prototypes of the International standard kilogramme supplied by the

International Bureau of Weights and Measures (BIPM) are available in many

other laboratories of different countries.

In India, this is available at the National Physical Laboratory (NPL), New

While dealing with atoms and molecules, the kilogramme is an inconvenient

unit. In this case, there is an important standard unit of mass, called the

unified atomic mass unit (u), which has been established for expressing the

mass of atoms as

1 unified atomic mass unit = 1 u

One unified mass unit is equal to (1/12) of the mass of an atom of Carbon-12

isotope (12C6 ) including the mass of electrons.

1 u = 1.66 × 10–27 kg

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23.
Methods of measuring mass

(i) By using a common balance.

(ii) Large masses in the Universe like planets, stars, etc.,

based on Newton’s law of gravitation can be measured by

using gravitational method.

(iii) For measurement of small masses of atomic/subatomic

particles etc., we make use of mass spectrograph in which

radius of the trajectory is proportional to the mass of a

charged particle moving in uniform electric and magnetic

field.

Range of Masses

The masses of the objects, we come across in the

Universe, vary over a very wide range.

These may vary from tiny mass of the order of 10-30 kg of

an electron to the huge mass of about 1055 kg of the

known Universe.

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(i) By using a common balance.

(ii) Large masses in the Universe like planets, stars, etc.,

based on Newton’s law of gravitation can be measured by

using gravitational method.

(iii) For measurement of small masses of atomic/subatomic

particles etc., we make use of mass spectrograph in which

radius of the trajectory is proportional to the mass of a

charged particle moving in uniform electric and magnetic

field.

Range of Masses

The masses of the objects, we come across in the

Universe, vary over a very wide range.

These may vary from tiny mass of the order of 10-30 kg of

an electron to the huge mass of about 1055 kg of the

known Universe.

Home Next Previous

24.
Range and Order of Masses (Mlargest : Msmallest = 1085 : 1 ≈ (1041)2)

S.No. Object Mass (kg)

1 Electron 10-30

2 Proton 10-27

3 Uranium atom 10-25

4 Red blood cell 10-13

5 Dust particle 10-9

6 Rain drop 10-6

7 Mosquito 10-5

8 Grape 10-3

9 Human 102

10 Automobile 103

11 Boeing 747 108

12 Moon 1023

13 Earth 1025

14 Sun 1030

15 Milky way galaxy 1041

16 Observable Universe 1055 Home Next Previous

S.No. Object Mass (kg)

1 Electron 10-30

2 Proton 10-27

3 Uranium atom 10-25

4 Red blood cell 10-13

5 Dust particle 10-9

6 Rain drop 10-6

7 Mosquito 10-5

8 Grape 10-3

9 Human 102

10 Automobile 103

11 Boeing 747 108

12 Moon 1023

13 Earth 1025

14 Sun 1030

15 Milky way galaxy 1041

16 Observable Universe 1055 Home Next Previous

25.
MEASUREMENT OF TIME

We use an atomic standard of time, which is based on the periodic

vibrations produced in a cesium atom. This is the basis of the cesium

clock, sometimes called atomic clock, used in the national standards.

In the cesium atomic clock, the second is taken as the time needed for

9,192,631,770 vibrations of the radiation corresponding to the transition

between the two hyperfine levels of the ground state of cesium-133 atom.

The vibrations of the cesium atom regulate the rate of this cesium atomic

clock just as the vibrations of a balance wheel regulate an ordinary

wristwatch or the vibrations of a small quartz crystal regulate a quartz

A cesium atomic clock is used at the National Physical Laboratory

(NPL), New Delhi to maintain the Indian standard of time.

Home Next Previous

We use an atomic standard of time, which is based on the periodic

vibrations produced in a cesium atom. This is the basis of the cesium

clock, sometimes called atomic clock, used in the national standards.

In the cesium atomic clock, the second is taken as the time needed for

9,192,631,770 vibrations of the radiation corresponding to the transition

between the two hyperfine levels of the ground state of cesium-133 atom.

The vibrations of the cesium atom regulate the rate of this cesium atomic

clock just as the vibrations of a balance wheel regulate an ordinary

wristwatch or the vibrations of a small quartz crystal regulate a quartz

A cesium atomic clock is used at the National Physical Laboratory

(NPL), New Delhi to maintain the Indian standard of time.

Home Next Previous

26.
Range and Order of Time Intervals (Tlongest : Tshortest = 1041 : 1)

S.No. Event Time Intervals (s)

1 Life span of most unstable particle 10-24

2 Time required for light to cross a nuclear distance 10-22

3 Period of X-rays 10-19

4 Period of atomic vibrations 10-15

5 Period of light wave 10-15

6 Life time of an excited atom 10 -8

7 Period of radio wave 10-6

8 Period of sound wave 10-3

9 Wink of eye 10-1

10 Time between successive human heart beats 100

11 Travel time for light from the Moon to the Earth 100

12 Travel time for light from the Sun to the Earth 102

13 Time period of a satellite 104

14 Rotation period of the Earth 105

15 Rotation and revolution periods of the Moon 106

16 Revolution period of the Earth 107

17 Travel time for light from the nearest star 108

18 Average human life span 109

19 Age of Egyptian pyramids 1011

20 Time since dinosaurs became extinct 1015

21 Age of the Universe 1017

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S.No. Event Time Intervals (s)

1 Life span of most unstable particle 10-24

2 Time required for light to cross a nuclear distance 10-22

3 Period of X-rays 10-19

4 Period of atomic vibrations 10-15

5 Period of light wave 10-15

6 Life time of an excited atom 10 -8

7 Period of radio wave 10-6

8 Period of sound wave 10-3

9 Wink of eye 10-1

10 Time between successive human heart beats 100

11 Travel time for light from the Moon to the Earth 100

12 Travel time for light from the Sun to the Earth 102

13 Time period of a satellite 104

14 Rotation period of the Earth 105

15 Rotation and revolution periods of the Moon 106

16 Revolution period of the Earth 107

17 Travel time for light from the nearest star 108

18 Average human life span 109

19 Age of Egyptian pyramids 1011

20 Time since dinosaurs became extinct 1015

21 Age of the Universe 1017

Home Next Previous

27.
ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN

The result of every measurement by any measuring instrument contains

some uncertainty. This uncertainty is called error.

The accuracy of a measurement is a measure of how close the measured

value is to the true value of the quantity.

Precision tells us to what resolution or limit the quantity is measured.

Suppose the true value of a certain length is near 2.874 cm.

In one experiment, using a measuring instrument of resolution 0.1 cm, the

measured value is found to be 2.7 cm, while in another experiment using a

measuring device of greater resolution, say 0.01 cm, the length is

determined to be 2.69 cm.

The first measurement has more accuracy (because it is closer to the true

value) but less precision (its resolution is only 0.1 cm), while the second

measurement is less accurate but more precise. Home Next Previous

The result of every measurement by any measuring instrument contains

some uncertainty. This uncertainty is called error.

The accuracy of a measurement is a measure of how close the measured

value is to the true value of the quantity.

Precision tells us to what resolution or limit the quantity is measured.

Suppose the true value of a certain length is near 2.874 cm.

In one experiment, using a measuring instrument of resolution 0.1 cm, the

measured value is found to be 2.7 cm, while in another experiment using a

measuring device of greater resolution, say 0.01 cm, the length is

determined to be 2.69 cm.

The first measurement has more accuracy (because it is closer to the true

value) but less precision (its resolution is only 0.1 cm), while the second

measurement is less accurate but more precise. Home Next Previous

28.
In general, the errors in measurement can be broadly classified as

(I) Systematic errors and (II) Random errors

I. Systematic errors

The systematic errors are those errors that tend to be in one direction, either

positive or negative.

Some of the sources of systematic errors are:

(a) Instrumental errors:

The instrumental errors that arise from the errors due to imperfect design

or calibration of the measuring instrument, zero error in the instrument,

etc.

(i)The temperature graduations of a thermometer may be inadequately

calibrated (it may read 104 °C at the boiling point of water at STP whereas it

should read 100 °C);

(ii)In a vernier callipers the zero mark of vernier scale may not coincide with

the zero mark of the main scale;

(iii) An ordinary metre scale may be worn off at one end. Home Next Previous

(I) Systematic errors and (II) Random errors

I. Systematic errors

The systematic errors are those errors that tend to be in one direction, either

positive or negative.

Some of the sources of systematic errors are:

(a) Instrumental errors:

The instrumental errors that arise from the errors due to imperfect design

or calibration of the measuring instrument, zero error in the instrument,

etc.

(i)The temperature graduations of a thermometer may be inadequately

calibrated (it may read 104 °C at the boiling point of water at STP whereas it

should read 100 °C);

(ii)In a vernier callipers the zero mark of vernier scale may not coincide with

the zero mark of the main scale;

(iii) An ordinary metre scale may be worn off at one end. Home Next Previous

29.
(b) Imperfection in experimental technique or procedure:

To determine the temperature of a human body, a thermometer placed

under the armpit will always give a temperature lower than the actual

value of the body temperature.

(c) Personal errors:

The personal errors arise due to an individual’s bias, lack of proper

setting of the apparatus or individual’s carelessness in taking

observations without observing proper precautions, etc.

If you hold your head a bit too far to the right while reading the

position of a needle on the scale, you will introduce an error due to

parallax.

Systematic errors can be minimized by

(i) improving experimental techniques,

(ii) selecting better instruments and

(iii) removing personal bias as far as possible.

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To determine the temperature of a human body, a thermometer placed

under the armpit will always give a temperature lower than the actual

value of the body temperature.

(c) Personal errors:

The personal errors arise due to an individual’s bias, lack of proper

setting of the apparatus or individual’s carelessness in taking

observations without observing proper precautions, etc.

If you hold your head a bit too far to the right while reading the

position of a needle on the scale, you will introduce an error due to

parallax.

Systematic errors can be minimized by

(i) improving experimental techniques,

(ii) selecting better instruments and

(iii) removing personal bias as far as possible.

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30.
II. Random errors

The random errors are those errors, which occur irregularly and hence are

random with respect to sign and size.

These can arise due to random and unpredictable fluctuations in

experimental conditions, personal errors by the observer taking readings,

When the same person repeats the same observation, it is very

likely that he may get different readings every time.

Least count error

Least count:

The smallest value that can be measured by the measuring instrument is

called its least count.

The least count error is the error associated with the resolution of the

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The random errors are those errors, which occur irregularly and hence are

random with respect to sign and size.

These can arise due to random and unpredictable fluctuations in

experimental conditions, personal errors by the observer taking readings,

When the same person repeats the same observation, it is very

likely that he may get different readings every time.

Least count error

Least count:

The smallest value that can be measured by the measuring instrument is

called its least count.

The least count error is the error associated with the resolution of the

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31.
(i) A Vernier callipers has the least count as 0.01 cm;

(ii) A spherometer may have a least count of 0.001 cm.

Using instruments of higher precision, improving experimental techniques,

etc., we can reduce the least count error.

Repeating the observations several times and taking the arithmetic mean of

all the observations, the mean value would be very close to the true value of

the measured quantity.

Note:

Least count error belongs to Random errors category but within a limited

size; it occurs with both systematic and random errors.

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(ii) A spherometer may have a least count of 0.001 cm.

Using instruments of higher precision, improving experimental techniques,

etc., we can reduce the least count error.

Repeating the observations several times and taking the arithmetic mean of

all the observations, the mean value would be very close to the true value of

the measured quantity.

Note:

Least count error belongs to Random errors category but within a limited

size; it occurs with both systematic and random errors.

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32.
Absolute Error, Relative Error and Percentage Error

Absolute error

The magnitude of the difference between the individual measurement

value and the true value of the quantity is called the absolute error of the

This is denoted by |Δa|.

Note: In absence of any other method of knowing true value, we consider

arithmetic mean as the true value.

The errors in the individual measurement values from the true value are:

Δa1 = a1 - amean

Δa2 = a2 - amean

----------------

----------------

Δan = an - amean

The Δa calculated above may be positive or negative.

But absolute error |Δa| will always be positive.

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Absolute error

The magnitude of the difference between the individual measurement

value and the true value of the quantity is called the absolute error of the

This is denoted by |Δa|.

Note: In absence of any other method of knowing true value, we consider

arithmetic mean as the true value.

The errors in the individual measurement values from the true value are:

Δa1 = a1 - amean

Δa2 = a2 - amean

----------------

----------------

Δan = an - amean

The Δa calculated above may be positive or negative.

But absolute error |Δa| will always be positive.

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33.
The arithmetic mean of all the absolute errors is taken as the final or mean

absolute error of the value of the physical quantity a.

It is represented by Δamean.

Δamean = (|Δa1|+|Δa2 |+|Δa3|+...+ |Δan|)/n

n

= ∑ |Δai|/n

i=1

If we do a single measurement, the value we get may be in the range

amean ± Δamean

This implies that any measurement of the physical quantity a is

likely to lie between

(amean + Δamean) and (amean - Δamean)

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absolute error of the value of the physical quantity a.

It is represented by Δamean.

Δamean = (|Δa1|+|Δa2 |+|Δa3|+...+ |Δan|)/n

n

= ∑ |Δai|/n

i=1

If we do a single measurement, the value we get may be in the range

amean ± Δamean

This implies that any measurement of the physical quantity a is

likely to lie between

(amean + Δamean) and (amean - Δamean)

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34.
Relative

The relative error is the ratio of the mean absolute error

Δamean to the mean value amean of the quantity measured.

Mean absolute error

Relative error =

True value or Arithmetic Mean

Δamean

Relative error =

amean

Percentage error

When the relative error is expressed in per cent, it is

called the percentage error (δa).

Mean absolute error

Percentage error = x 100%

True value or Arithmetic Mean

Δamean

Percentage error δa = x 100%

amean

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The relative error is the ratio of the mean absolute error

Δamean to the mean value amean of the quantity measured.

Mean absolute error

Relative error =

True value or Arithmetic Mean

Δamean

Relative error =

amean

Percentage error

When the relative error is expressed in per cent, it is

called the percentage error (δa).

Mean absolute error

Percentage error = x 100%

True value or Arithmetic Mean

Δamean

Percentage error δa = x 100%

amean

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35.
Combination of Errors

In an experiment involving several measurements, the errors in all the

measurements get combined.

Density is the ratio of the mass to the volume of the substance.

If there are errors in the measurement of mass and of the sizes or

dimensions, then there will be error in the density of the substance.

(a) Error of a Sum:

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=A+B

When two quantities

Z ± ΔZ = (A ± ΔA) + (B ± ΔB) are added, the

absolute error in

= (A + B) ± (ΔA + ΔB) the final result is the

sum of the

= Z ± (ΔA + ΔB) individual errors.

± ΔZ = ± (ΔA + ΔB) or ΔZ = (ΔA + ΔB)

Home Next Previous

In an experiment involving several measurements, the errors in all the

measurements get combined.

Density is the ratio of the mass to the volume of the substance.

If there are errors in the measurement of mass and of the sizes or

dimensions, then there will be error in the density of the substance.

(a) Error of a Sum:

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=A+B

When two quantities

Z ± ΔZ = (A ± ΔA) + (B ± ΔB) are added, the

absolute error in

= (A + B) ± (ΔA + ΔB) the final result is the

sum of the

= Z ± (ΔA + ΔB) individual errors.

± ΔZ = ± (ΔA + ΔB) or ΔZ = (ΔA + ΔB)

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36.
(b) Error of a Difference:

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=A-B

Z ± ΔZ = (A ± ΔA) - (B ± ΔB)

= (A - B) ± ΔA ΔB

±

= Z ± (ΔA + ΔB) (since ± and are the

±

same)

± ΔZ = ± (ΔA + ΔB) When two quantities

are subtracted, the

or ΔZ = (ΔA + ΔB) absolute error in

the final result is the

sum of the

individual errors.

Rule:

When two quantities are added or subtracted, the absolute error in the

final result is the sum of the absolute errors in the individual quantities.

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Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=A-B

Z ± ΔZ = (A ± ΔA) - (B ± ΔB)

= (A - B) ± ΔA ΔB

±

= Z ± (ΔA + ΔB) (since ± and are the

±

same)

± ΔZ = ± (ΔA + ΔB) When two quantities

are subtracted, the

or ΔZ = (ΔA + ΔB) absolute error in

the final result is the

sum of the

individual errors.

Rule:

When two quantities are added or subtracted, the absolute error in the

final result is the sum of the absolute errors in the individual quantities.

Home Next Previous

37.
(c) Error of a Product:

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=AxB

Z ± ΔZ = (A ± ΔA) x (B ± ΔB)

Z ± ΔZ = AB ± A ΔB ± B ΔA ± ΔA ΔB

Dividing LHS by Z and RHS by AB we have,

ΔZ ΔB ΔA ΔA ΔB

1± = 1± ± ±

Z B A AB

ΔZ ΔB ΔA ΔA ΔB

± = ± ± is very small and hence negligible

Z B A AB

When two quantities are

ΔZ ΔA ΔB multiplied, the relative error in

or = + the final result is the sum of the

Z A B relative errors of the individual

quantities.

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Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=AxB

Z ± ΔZ = (A ± ΔA) x (B ± ΔB)

Z ± ΔZ = AB ± A ΔB ± B ΔA ± ΔA ΔB

Dividing LHS by Z and RHS by AB we have,

ΔZ ΔB ΔA ΔA ΔB

1± = 1± ± ±

Z B A AB

ΔZ ΔB ΔA ΔA ΔB

± = ± ± is very small and hence negligible

Z B A AB

When two quantities are

ΔZ ΔA ΔB multiplied, the relative error in

or = + the final result is the sum of the

Z A B relative errors of the individual

quantities.

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38.
Error of a Product: ALITER

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=AxB

Applying log on both the sides, we have

log Z = log A + log B

Differentiating, we have

ΔZ ΔA ΔB

= +

Z A B

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Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z=AxB

Applying log on both the sides, we have

log Z = log A + log B

Differentiating, we have

ΔZ ΔA ΔB

= +

Z A B

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39.
(d) Error of a Quotient:

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

ΔB ΔA ΔA ΔB

Let Z = A Z ± ΔZ = A ± A x ± ± x

B B B B B B B

(A ± ΔA) Dividing LHS by Z and RHS by A / B and

Z ± ΔZ = simplifying we have,

(B ± ΔB)

(A ± ΔA) ΔZ ΔB ΔA

± = ± ±

Z ± ΔZ = Z B A

ΔB

B 1± ΔA ΔB

B is negligible

B2

(A ± ΔA) ΔB

-1 ΔZ ΔA ΔB

or = +

Z ± ΔZ = 1± Z A B

B B

When two quantities are

ΔA ΔB divided, the relative error in the

1

±

Z ± ΔZ = A ± final result is the sum of the

B B B

relative errors of the individual

(by Binomial Approximation) quantities. Home Next Previous

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

ΔB ΔA ΔA ΔB

Let Z = A Z ± ΔZ = A ± A x ± ± x

B B B B B B B

(A ± ΔA) Dividing LHS by Z and RHS by A / B and

Z ± ΔZ = simplifying we have,

(B ± ΔB)

(A ± ΔA) ΔZ ΔB ΔA

± = ± ±

Z ± ΔZ = Z B A

ΔB

B 1± ΔA ΔB

B is negligible

B2

(A ± ΔA) ΔB

-1 ΔZ ΔA ΔB

or = +

Z ± ΔZ = 1± Z A B

B B

When two quantities are

ΔA ΔB divided, the relative error in the

1

±

Z ± ΔZ = A ± final result is the sum of the

B B B

relative errors of the individual

(by Binomial Approximation) quantities. Home Next Previous

40.
Error of a Quotient: ALITER

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z = A

B

Applying log on both the sides, we have

log Z = log A - log B

Differentiating, we have

ΔZ ΔA ΔB

= -

Z A B

Logically an error can not be nullified by making another error. Therefore

errors are not subtracted but only added up.

ΔZ ΔB Math has to be bent to satisfy

ΔA

= + Physics in many situations!

Z A B Think of more such situations!!

When two quantities are multiplied or divided, the relative error in the

final result is the sum of the relative errors in the individual quantities.

Home Next Previous

Suppose two physical quantities A and B have measured values

A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.

Let Z = A

B

Applying log on both the sides, we have

log Z = log A - log B

Differentiating, we have

ΔZ ΔA ΔB

= -

Z A B

Logically an error can not be nullified by making another error. Therefore

errors are not subtracted but only added up.

ΔZ ΔB Math has to be bent to satisfy

ΔA

= + Physics in many situations!

Z A B Think of more such situations!!

When two quantities are multiplied or divided, the relative error in the

final result is the sum of the relative errors in the individual quantities.

Home Next Previous

41.
(e) Error of an Exponent (Power):

Suppose a physical quantity A has measured values A ± ΔA where ΔA is its

absolute error.

Let Z = Ap where p is a constant.

Z = A x A x A x ………x A (p times)

Z ± ΔZ = (A ± ΔA) x (A ± ΔA) x (A ± ΔA) x ……. x (A ± ΔA) (p times)

ΔZ ΔA ΔA ΔA ΔA

= + + + ……… + (p times as per the

Z A A A A

product rule for errors)

ΔZ ΔA Note:

or =p If p is negative, |p| is taken because errors

Z A due to multiple quantities get added up.

The relative error in a physical quantity raised to the power p is the

p times the relative error in the individual quantity.

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Suppose a physical quantity A has measured values A ± ΔA where ΔA is its

absolute error.

Let Z = Ap where p is a constant.

Z = A x A x A x ………x A (p times)

Z ± ΔZ = (A ± ΔA) x (A ± ΔA) x (A ± ΔA) x ……. x (A ± ΔA) (p times)

ΔZ ΔA ΔA ΔA ΔA

= + + + ……… + (p times as per the

Z A A A A

product rule for errors)

ΔZ ΔA Note:

or =p If p is negative, |p| is taken because errors

Z A due to multiple quantities get added up.

The relative error in a physical quantity raised to the power p is the

p times the relative error in the individual quantity.

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42.
(f) Error of an Exponent (Power): ALITER

Suppose a physical quantity A has measured values A ± ΔA where ΔA is its

absolute error.

Let Z = Ap where p is a constant.

Applying log on both the sides, we have

log Z = |p| log A (Whether p is positive or negative

errors due to multiple quantities get

Differentiating, we have added up only)

ΔZ ΔA

= |p|

Z A

Ap x Bq

In general, if Z = , then

C r

Note:

ΔZ ΔA ΔB ΔC

= p + q +r Cr is in Denominator, but the

Z A B C relative error is added up.

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Suppose a physical quantity A has measured values A ± ΔA where ΔA is its

absolute error.

Let Z = Ap where p is a constant.

Applying log on both the sides, we have

log Z = |p| log A (Whether p is positive or negative

errors due to multiple quantities get

Differentiating, we have added up only)

ΔZ ΔA

= |p|

Z A

Ap x Bq

In general, if Z = , then

C r

Note:

ΔZ ΔA ΔB ΔC

= p + q +r Cr is in Denominator, but the

Z A B C relative error is added up.

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43.
SIGNIFICANT FIGURES

The reported result of measurement is a number that includes all digits in

the number that are known reliably plus the first digit that is uncertain.

The reliable digits plus the first uncertain digit are known as significant

digits or significant figures.

(i) The period of oscillation of a simple pendulum is 2.36 s; the digits 2 and

3 are reliable and certain, while the digit 6 is uncertain. Thus, the measured

value has three significant figures.

(ii) The length of an object reported after measurement to be 287.5 cm has

four significant figures, the digits 2, 8, 7 are certain while the digit 5 is

A choice of change of different units does not change the number of

significant digits or figures in a measurement.

Eg. The length 1.205 cm, 0.01205, 12.05 mm and 12050 μm all have four SF.

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The reported result of measurement is a number that includes all digits in

the number that are known reliably plus the first digit that is uncertain.

The reliable digits plus the first uncertain digit are known as significant

digits or significant figures.

(i) The period of oscillation of a simple pendulum is 2.36 s; the digits 2 and

3 are reliable and certain, while the digit 6 is uncertain. Thus, the measured

value has three significant figures.

(ii) The length of an object reported after measurement to be 287.5 cm has

four significant figures, the digits 2, 8, 7 are certain while the digit 5 is

A choice of change of different units does not change the number of

significant digits or figures in a measurement.

Eg. The length 1.205 cm, 0.01205, 12.05 mm and 12050 μm all have four SF.

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44.
Rules for determining the number of significant figures

(i) All the non-zero digits are significant.

(ii) All the zeros between two non-zero digits are significant, no matter

where the decimal point is, if at all.

(iii) If the number is less than 1, the zero(s) on the right of decimal point but

to the left of the first non-zero digit are not significant.

(iv) The terminal or trailing zero(s) in a number without a decimal point are

not significant.

(v) The trailing zero(s) in a number with a decimal point are significant.

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(i) All the non-zero digits are significant.

(ii) All the zeros between two non-zero digits are significant, no matter

where the decimal point is, if at all.

(iii) If the number is less than 1, the zero(s) on the right of decimal point but

to the left of the first non-zero digit are not significant.

(iv) The terminal or trailing zero(s) in a number without a decimal point are

not significant.

(v) The trailing zero(s) in a number with a decimal point are significant.

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45.
Scientific Notation

Any given number can be written in the form of a×10b in many ways;

for example 350 can be written as 3.5×102 or 35×101 or 350×100.

a×10b means "a times ten raised to the power of b", where the exponent b is an

integer, and the coefficient a is any real number called the significand or

mantissa (the term "mantissa" is different from "mantissa" in common

If the number is negative then a minus sign precedes a (as in ordinary

decimal notation).

In normalized scientific notation, the exponent b is chosen such that the

absolute value of a remains at least one but less than ten (1 ≤ |a| < 10).

For example, 350 is written as 3.5×102.

This form allows easy comparison of two numbers of the same sign in a, as

the exponent b gives the number's order of magnitude.

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Any given number can be written in the form of a×10b in many ways;

for example 350 can be written as 3.5×102 or 35×101 or 350×100.

a×10b means "a times ten raised to the power of b", where the exponent b is an

integer, and the coefficient a is any real number called the significand or

mantissa (the term "mantissa" is different from "mantissa" in common

If the number is negative then a minus sign precedes a (as in ordinary

decimal notation).

In normalized scientific notation, the exponent b is chosen such that the

absolute value of a remains at least one but less than ten (1 ≤ |a| < 10).

For example, 350 is written as 3.5×102.

This form allows easy comparison of two numbers of the same sign in a, as

the exponent b gives the number's order of magnitude.

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46.
Rules for Arithmetic Operations with Significant Figures

In arithmetic operations the final result should not have more

significant figures than the original data from which it was

(1)Multiplication or division:

The final result should retain as many significant figures as are

there in the original number with the least significant figures.

(2) Addition or subtraction:

The final result should retain as many decimal places as are there

in the number with the least decimal places.

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In arithmetic operations the final result should not have more

significant figures than the original data from which it was

(1)Multiplication or division:

The final result should retain as many significant figures as are

there in the original number with the least significant figures.

(2) Addition or subtraction:

The final result should retain as many decimal places as are there

in the number with the least decimal places.

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47.
Rounding off the Uncertain Digits

Rounding off a number means dropping of digits which are not

significant. The following rules are followed for rounding off the

1.If the digits to be dropped are greater than five, then add one to the

preceding significant figure.

2. If the digit to be dropped is less than five then it is dropped without

bringing any change in the preceding significant figure.

3.If the digit to be dropped is five, then the preceding digit will be left

unchanged if the preceding digit is even and it will be increased by

one if it is odd.

4.In any involved or complex multi-step calculation, one should retain,

in intermediate steps, one digit more than the significant digits and

round off to proper significant figures at the end of the calculation.

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Rounding off a number means dropping of digits which are not

significant. The following rules are followed for rounding off the

1.If the digits to be dropped are greater than five, then add one to the

preceding significant figure.

2. If the digit to be dropped is less than five then it is dropped without

bringing any change in the preceding significant figure.

3.If the digit to be dropped is five, then the preceding digit will be left

unchanged if the preceding digit is even and it will be increased by

one if it is odd.

4.In any involved or complex multi-step calculation, one should retain,

in intermediate steps, one digit more than the significant digits and

round off to proper significant figures at the end of the calculation.

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48.
DIMENSIONS OF PHYSICAL QUANTITIES

The nature of a physical quantity is described by its dimensions.

All the physical quantities can be expressed in terms of the seven base or

fundamental quantities viz. mass, length, time, electric current,

thermodynamic temperature, intensity of light and amount of substance,

raised to some power.

The dimensions of a physical quantity are the powers (or exponents) to which

the fundamental or base quantities are raised to represent that quantity.

Using the square brackets [ ] around a quantity means that we are dealing

with ‘the dimensions of’ the quantity.

i)The dimensions of volume of an object are [L3]

ii)The dimensions of force are [MLT -2]

iii)The dimensions of energy are [ML2T-2]

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The nature of a physical quantity is described by its dimensions.

All the physical quantities can be expressed in terms of the seven base or

fundamental quantities viz. mass, length, time, electric current,

thermodynamic temperature, intensity of light and amount of substance,

raised to some power.

The dimensions of a physical quantity are the powers (or exponents) to which

the fundamental or base quantities are raised to represent that quantity.

Using the square brackets [ ] around a quantity means that we are dealing

with ‘the dimensions of’ the quantity.

i)The dimensions of volume of an object are [L3]

ii)The dimensions of force are [MLT -2]

iii)The dimensions of energy are [ML2T-2]

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49.
Dimensional Quantity

Dimensional quantity is a physical quantity which has dimensions.

For example: Speed, acceleration, momentum, torque, etc.

Dimensionless Quantity

Dimensionless quantity is a physical quantity which has no dimensions.

For example: Relative density, refractive index, strain, etc.

Dimensional Constant

Dimensional constant is a constant which has dimensions.

For example: Universal Gravitational constant, Planck’s constant, Hubble

constant, Stefan constant, Wien constant, Boltzmann constant, Universal

Gas constant, Faraday constant, etc.

Dimensionless Constant

Dimensionless constant is a constant which has no dimensions.

For example: 5, -.0.38, e, π, etc.

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Dimensional quantity is a physical quantity which has dimensions.

For example: Speed, acceleration, momentum, torque, etc.

Dimensionless Quantity

Dimensionless quantity is a physical quantity which has no dimensions.

For example: Relative density, refractive index, strain, etc.

Dimensional Constant

Dimensional constant is a constant which has dimensions.

For example: Universal Gravitational constant, Planck’s constant, Hubble

constant, Stefan constant, Wien constant, Boltzmann constant, Universal

Gas constant, Faraday constant, etc.

Dimensionless Constant

Dimensionless constant is a constant which has no dimensions.

For example: 5, -.0.38, e, π, etc.

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50.
DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS

The expression which shows how and which of the base quantities

represent the dimensions of a physical quantity is called the

dimensional formula of the given physical quantity.

(i)The dimensional formula of the volume is [M° L3 T°],

(ii)The dimensional formula of speed or velocity is [M° L T -1]

(iii) The dimensional formula of acceleration is [M° L T –2]

An equation obtained by equating a physical quantity with its

dimensional formula is called the dimensional equation of the physical

(i) [V] = [M° L3 T°]

(ii) [v] = [M° L T-1]

(iii) [a] = [M° L T–2]

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The expression which shows how and which of the base quantities

represent the dimensions of a physical quantity is called the

dimensional formula of the given physical quantity.

(i)The dimensional formula of the volume is [M° L3 T°],

(ii)The dimensional formula of speed or velocity is [M° L T -1]

(iii) The dimensional formula of acceleration is [M° L T –2]

An equation obtained by equating a physical quantity with its

dimensional formula is called the dimensional equation of the physical

(i) [V] = [M° L3 T°]

(ii) [v] = [M° L T-1]

(iii) [a] = [M° L T–2]

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51.
Quantities having the same dimensional formulae

1. Impulse and momentum

2. Work, energy, torque, moment of force

3. Angular momentum, Planck’s constant, rotational impulse

4. Stress, pressure, modulus of elasticity, energy density

5. Force constant, surface tension, surface energy

6. Angular velocity, frequency, velocity gradient

7. Gravitational potential, latent heat

8. Thermal capacity, entropy, universal gas constant and Boltzmann’s

const.

9. Force, thrust

10. Power, luminous flux

Dimensional formulae for physical quantities often used in Physics are

given at the end. (From Slide 63)

Home Next Previous

1. Impulse and momentum

2. Work, energy, torque, moment of force

3. Angular momentum, Planck’s constant, rotational impulse

4. Stress, pressure, modulus of elasticity, energy density

5. Force constant, surface tension, surface energy

6. Angular velocity, frequency, velocity gradient

7. Gravitational potential, latent heat

8. Thermal capacity, entropy, universal gas constant and Boltzmann’s

const.

9. Force, thrust

10. Power, luminous flux

Dimensional formulae for physical quantities often used in Physics are

given at the end. (From Slide 63)

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52.
DIMENSIONAL ANALYSIS AND ITS

Dimensional analysis is a tool to find or check relations among physical

quantities by using their dimensions.

When magnitudes of two or more physical quantities are multiplied, their

units should be treated in the same manner as ordinary algebraic symbols.

We can cancel identical units in the numerator and denominator.

Similarly, physical quantities represented by symbols on both sides of a

mathematical equation must have the same dimensions.

Dimensional Analysis can be used-

1.To check the dimensional consistency of equations

(Principle of homogeneity of dimensions).

2. To convert units in one system into another system.

3. To derive the relation between physical quantities based on certain

reasonable assumptions.

Home Next Previous

Dimensional analysis is a tool to find or check relations among physical

quantities by using their dimensions.

When magnitudes of two or more physical quantities are multiplied, their

units should be treated in the same manner as ordinary algebraic symbols.

We can cancel identical units in the numerator and denominator.

Similarly, physical quantities represented by symbols on both sides of a

mathematical equation must have the same dimensions.

Dimensional Analysis can be used-

1.To check the dimensional consistency of equations

(Principle of homogeneity of dimensions).

2. To convert units in one system into another system.

3. To derive the relation between physical quantities based on certain

reasonable assumptions.

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53.
I. Checking the Dimensional Consistency of Equations

The principle of homogeneity of dimensions:

The magnitudes of physical quantities may be added together or

subtracted from one another only if they have the same dimensions.

For example, initial velocity can be added to or subtracted from final

velocity because they have same dimensional formula [M0LT -1] .

But, force and momentum can not be added because their dimensional

formulae are different and are [MLT -2] and [MLT -1] respectively.

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The principle of homogeneity of dimensions:

The magnitudes of physical quantities may be added together or

subtracted from one another only if they have the same dimensions.

For example, initial velocity can be added to or subtracted from final

velocity because they have same dimensional formula [M0LT -1] .

But, force and momentum can not be added because their dimensional

formulae are different and are [MLT -2] and [MLT -1] respectively.

Home Next Previous

54.
1. To check the dimensional consistency of v2 = u2 + 2as

The dimensions of the quantities involved in the equation are:

[u] = [M0LT -1]

[v] = [M0LT -1]

[a] = [M0LT -2]

[s] = [M0LT 0]

Substituting the dimensions in the given equation,

[M0LT -1]2 = [M0LT -1]2 + [M0LT -2] [M0LT 0] (Note that the constant 2 in the term

‘2as’ does not have dimensions)

[M0L2T-2] = [M0L2T-2] + [M0L2T-2]

Each term of the above equation is having same dimensions.

Therefore, the given equation is dimensionally correct or dimensionally

consistent.

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The dimensions of the quantities involved in the equation are:

[u] = [M0LT -1]

[v] = [M0LT -1]

[a] = [M0LT -2]

[s] = [M0LT 0]

Substituting the dimensions in the given equation,

[M0LT -1]2 = [M0LT -1]2 + [M0LT -2] [M0LT 0] (Note that the constant 2 in the term

‘2as’ does not have dimensions)

[M0L2T-2] = [M0L2T-2] + [M0L2T-2]

Each term of the above equation is having same dimensions.

Therefore, the given equation is dimensionally correct or dimensionally

consistent.

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55.
If an equation fails the consistency test, it is proved wrong;

But if it passes, it is not proved right.

Thus, a dimensionally correct equation need not be actually an

exact (correct) equation, but a dimensionally wrong (incorrect) or

inconsistent equation must be wrong.

Example: Equations v2 = u2 - 2as or v2 = u2 + ½as are dimensionally

consistent but are incorrect equations in mechanics.

Albert Einstein tried his famous mass-energy equation as

E = m / c2, E = m2 / c, E = m2 c, etc.

Finally he settled with E = m c2 using dimensions and then proved it

with the help of Calculus.

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But if it passes, it is not proved right.

Thus, a dimensionally correct equation need not be actually an

exact (correct) equation, but a dimensionally wrong (incorrect) or

inconsistent equation must be wrong.

Example: Equations v2 = u2 - 2as or v2 = u2 + ½as are dimensionally

consistent but are incorrect equations in mechanics.

Albert Einstein tried his famous mass-energy equation as

E = m / c2, E = m2 / c, E = m2 c, etc.

Finally he settled with E = m c2 using dimensions and then proved it

with the help of Calculus.

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56.
2. To check the dimensional consistency of ½ mv2 = mgh

The dimensions of the quantities involved in the equation are:

[m] = [ML0T0]

[v] = [M0LT -1]

[g] = [M0LT -2]

[h] = [M0LT 0]

Substituting the dimensions in the given equation,

[ML0T0] [M0LT -1]2 = [ML0T0] [M0LT -2] [M0LT 0]

(Note that the constant ½ in the term

‘½ mv2 ’ does not have dimensions)

[ML2T-2] = [ML2T-2]

Each term of the above equation is having same dimensions.

Therefore, the given equation is dimensionally correct or dimensionally

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The dimensions of the quantities involved in the equation are:

[m] = [ML0T0]

[v] = [M0LT -1]

[g] = [M0LT -2]

[h] = [M0LT 0]

Substituting the dimensions in the given equation,

[ML0T0] [M0LT -1]2 = [ML0T0] [M0LT -2] [M0LT 0]

(Note that the constant ½ in the term

‘½ mv2 ’ does not have dimensions)

[ML2T-2] = [ML2T-2]

Each term of the above equation is having same dimensions.

Therefore, the given equation is dimensionally correct or dimensionally

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57.
II. Conversion of units in one system into another system

Units are derived from the dimensions and the dimensions are derived from

the actual formulae of physical quantities.

If the dimensions are known for a physical quantity, then it is easy to

express it in fps, cgs, mks, SI systems or any other arbitrary chosen system.

n1[M1aL1bT1c] = n2[M2aL2bT2c]

a b c

M1 L1 T1

n 2 = n1

M2 L2 T2

n1 and n2 are the magnitudes in the respective systems of units.

Smaller the unit bigger the magnitude of a physical quantity and vice versa.

For example, 1 m = 100 cm (m is the bigger unit and cm is the smaller one)

1 N = 105 dynes (Newton is bigger and dyne is smaller)

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Units are derived from the dimensions and the dimensions are derived from

the actual formulae of physical quantities.

If the dimensions are known for a physical quantity, then it is easy to

express it in fps, cgs, mks, SI systems or any other arbitrary chosen system.

n1[M1aL1bT1c] = n2[M2aL2bT2c]

a b c

M1 L1 T1

n 2 = n1

M2 L2 T2

n1 and n2 are the magnitudes in the respective systems of units.

Smaller the unit bigger the magnitude of a physical quantity and vice versa.

For example, 1 m = 100 cm (m is the bigger unit and cm is the smaller one)

1 N = 105 dynes (Newton is bigger and dyne is smaller)

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58.
1. To convert 1 joule in erg.

‘joule’ is unit of energy or work in SI system and ‘erg’ is the unit in cgs system.

The dimensional formula of energy or work is [ML2T-2].

The units from dimensions in SI and cgs systems are kg m2 s-2 and g cm2 s-2

Let n1 joule = n2 erg

SI System cgs System

Magnitude n1 = 1 n2 = ?

Mass (M) 1 kg (=1000 g) 1g

Length (L) 1 m (= 100 cm) 1 cm

Time (T) 1s 1s

[MaLbTc] = [ML2T-2] Therefore, a=1, b=2, c=-2

a b c n2 = 1 (1000)1 (100)2 (1)-2

M1 L1 T1

n2 = n1

M2 L2 T2 n2 = 107

1 2 -2

1000 g 100 cm 1s 1 joule = 107 erg

n2 = 1

1g 1 cm 1s Home Next Previous

‘joule’ is unit of energy or work in SI system and ‘erg’ is the unit in cgs system.

The dimensional formula of energy or work is [ML2T-2].

The units from dimensions in SI and cgs systems are kg m2 s-2 and g cm2 s-2

Let n1 joule = n2 erg

SI System cgs System

Magnitude n1 = 1 n2 = ?

Mass (M) 1 kg (=1000 g) 1g

Length (L) 1 m (= 100 cm) 1 cm

Time (T) 1s 1s

[MaLbTc] = [ML2T-2] Therefore, a=1, b=2, c=-2

a b c n2 = 1 (1000)1 (100)2 (1)-2

M1 L1 T1

n2 = n1

M2 L2 T2 n2 = 107

1 2 -2

1000 g 100 cm 1s 1 joule = 107 erg

n2 = 1

1g 1 cm 1s Home Next Previous

59.
2. To convert 1 newton into a system where mass is measured in

mg, length in km and time in minute

‘newton or kg m s-2’ is unit of force in SI system and ‘mg km min-2’ is the unit

in the new system.

The dimensional formula of force is [MLT -2].

Let n1 newton = n2 mg km min-2

SI System New System

Magnitude n1 = 1 n2 = ?

Mass (M) 1 kg (=106 mg) 1 mg

Length (L) 1 m (= 1/1000 km) 1 km

Time (T) 1 s (= 1/60) 1s

[MaLbTc] = [MLT -2] Therefore, a=1, b=1, c=-2

a b c

M1 L1 T1 n2 = 1 (106 )1 (10-3)1 (60)2

n2 = n1

M2 L2 T2 n2 = 3.6 x 106

1 1 -2

106 mg 1/1000 km 1/60 s

n2 = 1 1 newton =3.6x106mg km min-2

1 mg 1 km 1s

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mg, length in km and time in minute

‘newton or kg m s-2’ is unit of force in SI system and ‘mg km min-2’ is the unit

in the new system.

The dimensional formula of force is [MLT -2].

Let n1 newton = n2 mg km min-2

SI System New System

Magnitude n1 = 1 n2 = ?

Mass (M) 1 kg (=106 mg) 1 mg

Length (L) 1 m (= 1/1000 km) 1 km

Time (T) 1 s (= 1/60) 1s

[MaLbTc] = [MLT -2] Therefore, a=1, b=1, c=-2

a b c

M1 L1 T1 n2 = 1 (106 )1 (10-3)1 (60)2

n2 = n1

M2 L2 T2 n2 = 3.6 x 106

1 1 -2

106 mg 1/1000 km 1/60 s

n2 = 1 1 newton =3.6x106mg km min-2

1 mg 1 km 1s

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60.
III. Deducing Relation among the Physical Quantities

l

T =k

g The method of dimensions can sometimes be used to deduce relation

among the physical quantities.

For this we should know the dependence of the physical quantity on

other quantities (upto three physical quantities or linearly independent

variables) and consider it as a product type of the dependence.

Example:

1. Consider a simple pendulum, having a bob attached to a string that

oscillates under the action of the force of gravity. Suppose that the period

of oscillation of the simple pendulum depends on its length (l), mass of the

bob (m) and acceleration due to gravity (g). Derive the expression for its

time period using method of dimensions.

The dependence of time period T on the quantities l, g and m as a

product may be written as:

T = k l x my g z

where k is dimensionless constant and x, y and z are the exponents.

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l

T =k

g The method of dimensions can sometimes be used to deduce relation

among the physical quantities.

For this we should know the dependence of the physical quantity on

other quantities (upto three physical quantities or linearly independent

variables) and consider it as a product type of the dependence.

Example:

1. Consider a simple pendulum, having a bob attached to a string that

oscillates under the action of the force of gravity. Suppose that the period

of oscillation of the simple pendulum depends on its length (l), mass of the

bob (m) and acceleration due to gravity (g). Derive the expression for its

time period using method of dimensions.

The dependence of time period T on the quantities l, g and m as a

product may be written as:

T = k l x my g z

where k is dimensionless constant and x, y and z are the exponents.

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61.
The dimensions of the quantities involved in the equation are:

[m] = [ML0T0]

[l] = [M0LT 0]

[g] = [M0LT -2]

[T] = [M0L0T]

By substituting dimensions on both sides of T = k l x my gz, we have

[M0L0T] = [M0LT 0]x [ML0T0]y [M0LT -2]z

[M0L0T] = [M]y [L]x+z [T]-2z

On equating the dimensions on both sides, we have

y=0

x+z=0

–2z = 1

So that x = ½ , y = 0, z = -½

Then, T = k l½ g–½

l The value of k is 2π l

Or T = k and T = 2π

g g

determined from

other Home Next Previous

[m] = [ML0T0]

[l] = [M0LT 0]

[g] = [M0LT -2]

[T] = [M0L0T]

By substituting dimensions on both sides of T = k l x my gz, we have

[M0L0T] = [M0LT 0]x [ML0T0]y [M0LT -2]z

[M0L0T] = [M]y [L]x+z [T]-2z

On equating the dimensions on both sides, we have

y=0

x+z=0

–2z = 1

So that x = ½ , y = 0, z = -½

Then, T = k l½ g–½

l The value of k is 2π l

Or T = k and T = 2π

g g

determined from

other Home Next Previous

62.
Demerits of Dimensional Analysis

The dimensional analysis can not be used in the following cases:

1.The value of constants in an equation can not be determined as the

constants do not have dimensions.

2.Only dimensional consistency and not the physical consistency can be

3.Dimensions can be found from the physical quantity, but physical

quantity can not be always guessed from dimensions because two or more

quantities may have same dimensions.

4.The equation containing the dependency on more than 3 quantities can

not be determined using only M, L and T.

(Note that if 4 independent quantities are involved, then 4 variables

and hence 4 simultaneous equations are required; hence there must

be 4 fundamental dimensions)

5. The equation containing exponential, trigonometric, logarithmic

functions, etc. can not be derived as they do not have dimensions.

6.The equations having the relations other than products / quotients can

not be derived. Home Next Previous

The dimensional analysis can not be used in the following cases:

1.The value of constants in an equation can not be determined as the

constants do not have dimensions.

2.Only dimensional consistency and not the physical consistency can be

3.Dimensions can be found from the physical quantity, but physical

quantity can not be always guessed from dimensions because two or more

quantities may have same dimensions.

4.The equation containing the dependency on more than 3 quantities can

not be determined using only M, L and T.

(Note that if 4 independent quantities are involved, then 4 variables

and hence 4 simultaneous equations are required; hence there must

be 4 fundamental dimensions)

5. The equation containing exponential, trigonometric, logarithmic

functions, etc. can not be derived as they do not have dimensions.

6.The equations having the relations other than products / quotients can

not be derived. Home Next Previous

63.
Dimensional formulae for some physical quantities

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64.
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65.
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66.
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67.
V

V

dv

dx

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V

dv

dx

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68.
change in temperature

distance

S Q / T

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distance

S Q / T

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69.
heat energy

area x time x temperature 4

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area x time x temperature 4

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70.
potential difference

current

Q 1Q 2

o

4Fd 2

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current

Q 1Q 2

o

4Fd 2

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71.
4Fd 2

o

m1m 2

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o

m1m 2

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72.
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73.
]

Acknowledg

ement

Physics Part I for Class XI by NCERT

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Acknowledg

ement

Physics Part I for Class XI by NCERT

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