UNITS AND MEASUREMENT
Fundamental Quantity
A quantity which is measurable is called ‘physical quantity’.
Examples: Length, Mass, Time, etc.
A physical quantity which can be derived or expressed from base or fundamental quantity / quantities is called ‘derived quantity’.
Physical Quantity
A physical quantity which is the base and can not be derived from any other quantity is called ‘fundamental quantity’.
Examples: Speed, velocity, acceleration, force, momentum, torque, energy, pressure, density, thermal conductivity, resistance, magnetic moment, etc.
Derived Quantity
Unit
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.
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 requirements:
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.
Fundamental Units in SI system
| Quantity | Symbol | SI unit | Symbol |
Main units | Length | L | metre | m |
Mass | M | kilogramme | kg | |
Time | T | second | s | |
Electric Current | A | ampere | A | |
Thermodynamic Temperature | K | kelvin | K | |
Light Intensity | Cd | candela | cd | |
Amount of substance | mole | mole | mol | |
Supplementary units | Plane angle | dθ | radian | rad |
Solid angle | dΩ | steradian | sr |
Metre
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)
Kilogramme
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)
Second
Ampere
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)
Definition of SI Units
Kelvin
The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967)
Candela
Mole
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)
Plane angle
Plane angle ‘dθ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’.
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’.
dθ
ds
r
r
O
r
dΩ
r
dA
O
dθ =
ds
r
dΩ =
dA
r2
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 capital
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.
IMPORTANT
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 pressure | atm | 101325 Pa = 1.013 x 105 Pa |
Some Units are retained for general use (Though outside SI)
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.
Example:
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.
P
Measurement of Large Distances
The following indirect methods may be used to measure very large distances:
D
D
R
L
b
θ
θ =
LR
D
b
D
=
D =
b
θ
d
If ‘d’ is the diameter of the planet and ‘α’ is 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 location on the earth.
It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope.
Since D is known, the diameter d of the planet can be determined from
Measurement of the size or angular diameter of an astronomical object
α
D
D
d = α D
E
Echo method or Reflection method
S = the distance of hill from the observer
and
T = total time taken, then
Gun fire
Echo received
Sound wave
S
S
S =
v x T
2
where c = 3 x 108 m s-1
S =
c x t
2
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 fields.
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.
Estimation of Very Small Distances
1. Using Electron Microscope:
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.
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.
2. Avogadro’s Method:
20 x 20
1
acid per cm3 of solution.
cm3 of oleic
So, the concentration of the solution is
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
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
or
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.
20 x 20
1
nV
cm3
Amount of oleic acid in this solution =
t =
Volume of the film
Area of the film
t =
20 x 20 x A
nV
cm
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:
Range of Lengths
Unit | Symbol | Value | Definition |
1 fermi | 1 f | 10–15 m | |
1 angstrom | 1 Å | 10–10 m | |
1 Astronomical Unit | 1 AU | 1.496 × 1011 m | Average distance of the Sun from the Earth |
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 (3.26 ly) | The distance at which average radius of Earth’s orbit subtends an angle of 1 arc second |
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 |
Range and Order of Lengths
(Llongest : Lshortest = 1041 : 1)
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 Delhi.
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.
MEASUREMENT OF MASS
1 u = 1.66 × 10–27 kg
based on Newton’s law of gravitation can be measured by
using gravitational method.
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.
Methods of measuring mass
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.
Range of Masses
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 |
Range and Order of Masses
(Mlargest : Msmallest = 1085 : 1 ≈ (1041)2)
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 wristwatch.
MEASUREMENT OF TIME
A cesium atomic clock is used at the National Physical Laboratory (NPL), New Delhi to maintain the Indian standard of time.
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 |
Range and Order of Time Intervals
(Tlongest : Tshortest = 1041 : 1)
Error:
The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.
Accuracy:
The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.
Precision:
Precision tells us to what resolution or limit the quantity is measured.
Example:
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.
ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT
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.
Example:
the zero mark of the main scale;
(iii) An ordinary metre scale may be worn off at one end.
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.
Example:
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
(b) Imperfection in experimental technique or procedure:
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, etc.
Example:
When the same person repeats the same observation, it is very
likely that he may get different readings every time.
II. Random errors
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 instrument.
Example:
(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.
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 measurement.
This is denoted by |Δa|.
Note: In absence of any other method of knowing true value, we consider arithmetic mean as the true value.
Absolute Error, Relative Error and Percentage Error
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.
----------------
----------------
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.
Thus,
Δamean = (|Δa1|+|Δa2 |+|Δa3|+...+ |Δan|)/n
= ∑
|Δai|/n
i=1
n
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)
The relative error is the ratio of the mean absolute error
Δamean to the mean value amean of the quantity measured.
When the relative error is expressed in per cent, it is
called the percentage error (δa).
Relative error
Relative error =
Δamean
amean
Relative error =
Mean absolute error
True value or Arithmetic Mean
Percentage error
Percentage error =
Mean absolute error
True value or Arithmetic Mean
x 100%
Percentage error δa =
Δamean
amean
x 100%
In an experiment involving several measurements, the errors in all the measurements get combined.�
Example:
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.
Combination of Errors
Let Z = A + B
Z ± ΔZ = (A ± ΔA) + (B ± ΔB)
= (A + B) ± (ΔA + ΔB)
= Z ± (ΔA + ΔB)
± ΔZ = ± (ΔA + ΔB)
When two quantities are added, the absolute error in the final result is the sum of the individual errors.
or
ΔZ = (ΔA + ΔB)
(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)
When two quantities are subtracted, the absolute error in the final result is the sum of the individual errors.
ΔZ = (ΔA + ΔB)
±
= (A - B) ± ΔA ΔB
= Z ± (ΔA + ΔB) (since ± and are the same)
±
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.
or
± ΔZ = ± (ΔA + ΔB)
(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 = A x B
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,
1 ±
ΔZ
Z
=
ΔB
B
1 ±
ΔA
A
±
±
ΔA ΔB
A B
ΔA ΔB
A B
is very small and hence negligible
±
ΔZ
Z
=
ΔB
B
±
ΔA
A
±
or
=
ΔZ
Z
ΔA
A
ΔB
B
+
When two quantities are multiplied, the relative error in the final result is the sum of the relative errors of the individual quantities.
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 = A x B
Applying log on both the sides, we have
log Z = log A + log B
Differentiating, we have
=
ΔZ
Z
ΔA
A
ΔB
B
+
(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.
Let
Z =
A
B
(A ± ΔA)
(B ± ΔB)
Z ± ΔZ =
(A ± ΔA)
Z ± ΔZ =
B
ΔB
B
1 ±
Z ± ΔZ =
(A ± ΔA)
B
ΔB
B
1 ±
-1
(by Binomial Approximation)
=
A
B
ΔA
B
±
Z ± ΔZ
ΔB
B
1
±
ΔB
B
x
A
B
ΔA
B
±
Z ± ΔZ =
A
B
±
±
ΔB
B
x
ΔA
B
Dividing LHS by Z and RHS by A / B and simplifying we have,
±
ΔZ
Z
=
ΔB
B
±
ΔA
A
±
ΔA ΔB
B2
is negligible
or
=
ΔZ
Z
ΔA
A
ΔB
B
+
When two quantities are divided, the relative error in the final result is the sum of the relative errors of the individual quantities.
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
Z
ΔA
A
ΔB
B
-
Logically an error can not be nullified by making another error. Therefore errors are not subtracted but only added up.
=
ΔZ
Z
ΔA
A
ΔB
B
+
Math has to be bent to satisfy Physics in many situations!
Think of more such situations!!
Rule:
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.
(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 ± ΔZ = (A ± ΔA) x (A ± ΔA) x (A ± ΔA) x ……. x (A ± ΔA) (p times)
Z = A x A x A x ………x A (p times)
or
=
ΔZ
Z
ΔA
A
ΔA
A
+
ΔA
A
+
ΔA
A
+
+ ………
(p times as per the
product rule for errors)
=
ΔZ
Z
ΔA
A
p
Rule:
The relative error in a physical quantity raised to the power p is the p times the relative error in the individual quantity.
Note:
If p is negative, |p| is taken because errors due to multiple quantities get added up.
Let Z = Ap where p is a constant.
(f) Error of an Exponent (Power): ALITER
Suppose a physical quantity A has measured values A ± ΔA where ΔA is its absolute error.
Applying log on both the sides, we have
log Z = |p| log A
Differentiating, we have
=
ΔZ
Z
|p|
ΔA
A
=
ΔZ
Z
ΔA
A
p
+
ΔB
B
q
ΔC
C
r
+
In general, if , then
Z =
Ap x Bq
Cr
(Whether p is positive or negative errors due to multiple quantities get added up only)
Note:
Cr is in Denominator, but the relative error is added up.
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.
Example:
(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 uncertain.
SIGNIFICANT FIGURES
Note:
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.
where the decimal point is, if at all.
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.
Rules for determining the number of significant figures
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 logarithm).
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.
Scientific Notation
In arithmetic operations the final result should not have more significant figures than the original data from which it was obtained.
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.
Rules for Arithmetic Operations with Significant Figures
Rounding off a number means dropping of digits which are not significant. The following rules are followed for rounding off the number:
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.
unchanged if the preceding digit is even and it will be increased by
one if it is odd.
Rounding off the Uncertain Digits
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.
Note:
Using the square brackets [ ] around a quantity means that we are dealing with ‘the dimensions of’ the quantity.
Example:
DIMENSIONS OF PHYSICAL QUANTITIES
Dimensional Quantity
Dimensional quantity is a physical quantity which has dimensions.
For example: Speed, acceleration, momentum, torque, etc.
Dimensionless quantity is a physical quantity which has no dimensions.
For example: Relative density, refractive index, strain, etc.
Dimensionless Quantity
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 is a constant which has no dimensions.
For example: 5, -.0.38, e, π, etc.
Dimensionless Constant
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.
Example:
(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 quantity.
DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS
Example:
(i) [V] = [M° L3 T°]
(ii) [v] = [M° L T-1]
(iii) [a] = [M° L T–2]
Quantities having the same dimensional formulae
Dimensional formulae for physical quantities often used in Physics are given at the end. (From Slide 63)
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 AND ITS APPLICATIONS
Dimensional Analysis can be used-
(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.
Dimensional analysis is a tool to find or check relations among physical quantities by using their 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.
I. Checking the Dimensional Consistency of Equations
The principle of homogeneity of dimensions:
Example:
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] = [M0LT0]
Substituting the dimensions in the given equation,
[M0LT-1]2 = [M0LT-1]2 + [M0LT-2] [M0LT0]
[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.
(Note that the constant 2 in the term ‘2as’ does not have dimensions)
Note:
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.
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] = [M0LT0]
Substituting the dimensions in the given equation,
[ML0T0] [M0LT-1]2 = [ML0T0] [M0LT-2] [M0LT0]
[ML2T-2] = [ML2T-2]
Each term of the above equation is having same dimensions.
Therefore, the given equation is dimensionally correct or dimensionally consistent.
(Note that the constant ½ in the term ‘½ mv2 ’ does not have dimensions)
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]
M1
M2
T1
T2
L1
L2
a b c
n2 = n1
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)
Example:
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 respectively.
| SI System | cgs System |
Magnitude | n1 = 1 | n2 = ? |
Mass (M) | 1 kg (=1000 g) | 1 g |
Length (L) | 1 m (= 100 cm) | 1 cm |
Time (T) | 1 s | 1 s |
[MaLbTc] = [ML2T-2] Therefore, a=1, b=2, c=-2 | ||
M1
M2
T1
T2
L1
L2
a b c
n2 = n1
1000 g
1 g
1 2 -2
n2 = 1
100 cm
1 cm
1 s
1 s
n2 = 1 (1000)1 (100)2 (1)-2
n2 = 107
1 joule = 107 erg
Let n1 joule = n2 erg
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].
| 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) | 1 s |
[MaLbTc] = [MLT-2] Therefore, a=1, b=1, c=-2 | ||
M1
M2
T1
T2
L1
L2
a b c
n2 = n1
106 mg
1 mg
1 1 -2
n2 = 1
1/1000 km
1 km
1/60 s
1 s
n2 = 1 (106 )1 (10-3)1 (60)2
n2 = 3.6 x 106
1 newton =3.6x106mg km min-2
Let n1 newton = n2 mg km min-2
III. Deducing Relation among the Physical Quantities
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 lx my gz
where k is dimensionless constant and x, y and z are the exponents.
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:
By substituting dimensions on both sides of T = k lx my gz, we have
[M0L0T] = [M0LT0]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
Or
The dimensions of the quantities involved in the equation are:
[m] = [ML0T0]
[l] = [M0LT0]
[g] = [M0LT-2]
[T] = [M0L0T]
T = k
l
g
The value of k is 2π and
determined from other
methods.
T = 2π
l
g
Demerits of Dimensional Analysis
The dimensional analysis can not be used in the following cases:
(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.