Introduction
The international system of units
Measurement of length
Measurement of mass
Measurement of time
Accuracy, precision of instruments and errors in measurement
Significant figures
Dimensions of physical quantities
Dimensional formulae and dimensional equations
Dimensional analysis and its applications
INTRODUCTION
Measurement of any physical quantity involves comparison with a certain basic, arbitrarily
chosen, internationally accepted reference standard called unit. The result
of a measurement of a physical quantity is expressed by a number (or numerical
measure) accompanied by a unit. Although the number of physical quantities appears
to be very large, we need only a limited number of units for expressing all the
physical quantities, since they are interrelated with one another. The units for the
fundamental or base quantities are called fundamental or base units. The units of
all other physical quantities can be expressed as combinations of the base units.
Such units obtained for the derived quantities are called derived units. A complete
set of these units, both the base units and derived units, is known as the system of
units.
THE INTERNATIONAL SYSTEM OF UNITS
Three systems of units, the CGS, the FPS (or British) system and the MKS system
were in use extensively till recently. The base units for length, mass and time in
these systems were as follows :
• In CGS system they were centimetre, gram and second respectively.
• In FPS system they were foot, pound and second respectively.
• In MKS system they were metre, kilogram and second respectively.
The system of units which is at present internationally accepted for measurement is
the Système Internationale d’ Unites (French for International System of Units), abbreviated
as SI. The SI, 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. Because SI units used decimal system, conversions within the system are
quite simple and convenient.
BasequantitySI Units
NameSymbolDefinitionLengthMetreMThe metre is the length of the path travelledby light in vacuum during a time interval of1/299,792,458 of a second. (1983)MassKilogramKgThe kilogram is equal to the mass of the internationalprototype of the kilogram (a platinum-iridium alloy cylinder) kept at internationalBureau of Weights and Measures, atSevres, near Paris, France. (1889)TimeSecondSecThe second is the duration of 9,192,631,770periods of the radiation corresponding to thetransition between the two hyperfine levels ofthe ground state of the cesium-133 atom.(1967)ElectricAmpereAThe ampere is that constant current which, ifmaintained in current two straight parallelconductors of infinite length, of negligible circularcross-section, and placed 1 metre apartin vacuum,would produce between these conductors aforce equal to 2×10–7 newton per metre oflength. (1948ThermodynamicTemperatureKelvinKThe kelvin, is the fraction 1/273.16 of thedynamic temperature of the triple point ofwater. (1967)Amount of SubstanceMoleMolThe mole is the amount of substance of asystem, which contains as many elementaryentities as there are atoms in 0.012 kilogramof carbon - 12. (1971)LuminousIntensityCandelaCdThe candela is the luminous intensity, in agiven direction, of a source that emits monochromaticradiation of frequency 540×1012hertz and that has a radiant intensity in that direction of 1/683 watt pesteradian(1979)
Base SI Units
quantity Name Symbol Definition
Length Metre M 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)
Mass Kilogram Kg 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)
Time Second Sec 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)
Electric Ampere A The ampere is that constant current which, if
maintained in current 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.
Thermodynamic Kelvin K The kelvin, is the fraction 1/273.16 of the
Temperature dynamic temperature of the triple point of water.
Amount of Substance
Mole
Mol
The mole is the amount of substance of a
system, which contains as many elementary
entities as there are atoms in 0.012 kilogram
of carbon - 12. (1971)
Luminous
Intensity
Candela
Cd
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 pesteradian(1979)
w
Some units retained for general use (Though outside SI)
Name Symbol Value in SI units
Minute min 60s
Hour h 60 min=3600s
Day d 24 h=86400s
Year y 365.25 d=3.156 X 107s
Degree ˚ 1˚=(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 2.58X10-4 C/kg
Quintal q 100 kg
Barn b 100fm² = 10-25 m²
Are a 1 dam² = 10²m²
Hectare ha 1 hm² = 104 m²
Standard atmospheric pressure atm 101325 Pa = 1.013 X 105 Pa
MEASUREMENT OF LENGTH
Measuring Smaller Lengths: A metre scale is used for lengths from 10–3 m to 102 m.
A vernier callipers is used for lengths to an accuracy of 10–4 m. A screw gauge and a
spherometer can be used to measure lengths as less as to 10–5 m. To measure
lengths beyond these ranges, we make use of some special indirect methods.
Measurement of Large Distances
Large distances such as the distance of a planet or a star from the earth cannot be
measured directly with a metre scale. An important method in such cases is the parallax
method. When you hold a pencil in front of you against some specific point on
the background (a wall) and look at the pencil first through your left eye A (closing
the right eye) and then look at the pencil through your right eye B (closing the left
eye), you would notice that the position of the pencil seems to change with respect
to the point on the wall. This is called parallax. The distance between the two points
of observation is called the basis. In this example, the basis is the distance between
the eyes.
To measure the distance D of a far away planet S by the parallax method, we observe
it from two different positions (observatories) A and B on the Earth, separated
by distance AB = b at the same time as shown in the given figure. We measure the
angle between the two directions along which the planet is viewed at these two
points. The ° Ð ASB in the figure represented by symbol θ is called the parallax
angle or parallactic angle.
As the planet is very far away,
D
b
<<1
and therefore, θ is very small. Then we approximately take AB as an arc of length b
of a circle with centre at S and the distance D as the radius AS = BS
so that AB = b = D θ where θ is in radians.
D =
q
b
……………………………….. (1)
Having determined D, we can employ a similar method to determine the size or angular
diameter of the planet. If d is the diameter of the planet and α the angular size
of the planet (the angle subtended by d at the earth),
we have
a = d/D ………………………. (2)
The angle a 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 using Equation (2).
Example:
Calculate the angle of
(a) 1˚ (degree)
(b) 1′ (minute of arc or arcmin) and
(c) 1″ (second of arc or arc second) in radians.
Use 3600=2p rad, 1˚=60′ and 1′ = 60 ″
Answer:
(a) We have 3600 = 2π rad
1˚ = (
180
p
) rad = 1.745×10–2 rad
(b) 1˚ = 60′ = 1.745×10–2 rad
Hence, 1′ = 1.745´ 10-2 ¸ 60
= 2.908×10–4 rad (can be rounded up to 2.91×10–4 rad )
(c) 1′ = 60″ = 2.908×10–4 rad
Hence, 1″ = 2.908´ 10-4 ¸ 60
= 4.847×10–6 rad ( can be rounded up to 4.85×10–6 rad )
Example: A man wishes to estimate the distance of a nearby tower from him. He
stands at a point A in front of the tower C and spots a very distant object O in line
with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks
at O and C again. Since O is very distant, the direction BO is practically the same as
AO; but he finds the line of sight of C shifted from the original line of sight by an
angle θ = 40˚ ( θ is known as ‘parallax’) estimate the distance of the tower C from
his original position A.
Answer: We have, parallax angle θ = 40˚
From the figure,
AB = AC tan θ
AC =
tanq
AB
=
tan 40°
100m
= 100 m/0.8391 = 119 m
Example: The moon is observed from two diametrically opposite points A and B on
Earth. The angle θ subtended at the moon by the two directions of observation is 1˚
54′. Given the diameter of the Earth to be about 1.276 ´ 107 m, compute the distance
of the moon from the Earth.
Answer: We have θ = 1° 54′ = 114′
=(114´ 60”)´ (4.85 ´ 10-6) rad
= 3.32 ´ 10-2 rad,
since, 1”= 4.85 ´ 10-6 rad
Also b = AB =1.276 ´ 107
Hence from Equation (1), we have the earth-moon
distance,
D=
q
b
=1.276 ´ 107 ¸ 3.32 ´ 10-2
=3.84´ 108 m
Example: The Sun’s angular diameter is measured to be 1920”. The distance D of
the Sun from the Earth is 1.496 × 1011 m. What is the diameter of the Sun ?
Fig. 2.2 Parallax method.
PHYSICS 20
�
Answer Sun’s angular diameter
a = 1920"
=1920´ (4.85 ´ 10-6) rad
= 9.31´ 10-3
Sun’s diameter
d = a D
= 9.31´ 10-3 ´ 1.496 × 1011
= 1.39´ 109 m
Estimation of Very Small Distances: Size of a Molecule
To measure a very small size like that of a molecule (10–8 m to 10–10 m), we have to
adopt special methods. We cannot use a screw gauge or similar instruments. Even a
microscope has certain limitations.
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. Then we take 1 cm3 of this solution
and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is equal
to ÷ø
ö çèæ
20´ 20
1
cm3 of oleic acid/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.
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
Amount of oleic acid in this solution
= nV ÷ø
ö çè
æ
20´ 20
1
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
t =
Area of the film
Volume of the film
or,
A
t nV
20´ 20
= cm …………………………. (3)
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 10–9 m.
Example: If the size of a nucleus (in the range of 10–15 to 10–14 m) is scaled up
to the tip of a sharp pin, what roughly is the size of an atom ? Assume tip of the pin
to be in the range 10–5m to 10–4m.
Answer: The size of a nucleus is in the range of 10–15 m and 10–14 m. The tip of a
sharp pin is taken to be in the range of 10–5 m and 10–4 m. Thus we are scaling up by
a factor of 1010. An atom roughly of size 10–10 m will be scaled up to a size of 1 m.
Thus a nucleus in an atom is as small in size as the tip of a sharp pin placed at the
centre of a sphere of radius about a metre long.
MEASUREMENT OF MASS
Range of Lengths
There are certain special length units for short and large lengths. These
are
1 fermi = 1 f = 10–15 m
1 angstrom = 1 Å = 10–10 m
1 astronomical unit = 1 AU (average distance of the Sun from the Earth)
= 1.496´ 1011 m
1 light year = 1 ly= 9.46 ´ 1015 m
(distance that light travels with velocity of 3 ´ 108 m s–1 in 1 year)
1 parsec = 3.08 ´ 1016 m
(Parsec is the distance at which average radius of earth’s orbit subtends an
angle of 1 arc second)
Mass is a basic property of matter. It does not depend on the temperature, pressure
or location of the object in space. The SI unit of mass is kilogram (kg). The prototypes
of the International standard kilogram 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 kilogram 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 = 1u
= (1/12) of the mass of an atom of carbon-12
isotope (
12
6 C) including the mass of electrons
= 1.66 × 10–27 kg
Mass of commonly available objects can be determined by a common balance like
the one used in a grocery shop. Large masses in the universe like planets, stars,
etc., based on Newton’s law of gravitation can be measured by using gravitational
method. 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.
MEASUREMENT OF TIME
To measure any time interval we need a clock. We now 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. Such standards are available in many laboratories. 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.
The cesium atomic clocks are very accurate. In principle they provide portable standard.
The national standard of time interval ‘second’ as well as the frequency is maintained
through four cesium atomic clocks. A cesium atomic clock is used at the National
Physical Laboratory (NPL), New Delhi to maintain the Indian standard of time.
In our country, the NPL has the responsibility of maintenance and improvement of
physical standards, including that of time, frequency, etc. Note that the Indian
Standard Time (IST) is linked to this set of atomic clocks. The efficient cesium atomic
clocks are so accurate that they impart the uncertainty in time realisation as
± 1´ 10–13,i.e. 1 part in 1013.
This implies that the uncertainty gained over time by such a device is less than 1
part in 1013; they lose or gain no more than 3 μs in one year. In view of the trewww.
mendous accuracy in time measurement, the SI unit of length has been expressed in
terms the path length light travels in certain interval of time (1/299,792,458 of a
second).
ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT
Measurement is the foundation of all experimental science and technology. The result
of every measurement by any measuring instrument contains some uncertainty. This
uncertainty is called error. Every calculated quantity which is based on measured
values, also has an error. We shall distinguish between two terms: accuracy and
precision. 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.
The accuracy in measurement may depend on several factors, including the limit or
the resolution of the measuring instrument. For example, suppose the true value of a
certain length is near 3.678 cm. In one experiment, using a measuring instrument of
resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment
using a measuring device of greater resolution, say 0.01 cm, the length is determined
to be 3.38 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. Thus every measurement is
approximate due to errors in measurement.
In general, the errors in measurement can be broadly classified as (a) systematic errors
and (b) random errors.
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 that arise from the errors due to imperfect design or calibration
of the measuring instrument, zero error in the instrument, etc. For example,
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);
in a vernier callipers the zero mark of vernier scale may not coincide with the zero
mark of the main scale, or simply an ordinary metre scale may be worn off at one
end.
(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. Other external
conditions (such as changes in temperature, humidity, wind velocity, etc.) during
the experiment may systematically affect the measurement.
(c) Personal errors that 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. For example, if you, by habit, always 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 minimised by improving experimental techniques, selecting
better instruments and removing personal bias as far as possible. For a given set-up,
these errors may be estimated to a certain extent and the necessary corrections may
be applied to the readings.
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 (e.g. unpredictable fluctuations
in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc),
personal (unbiased) errors by the observer taking readings, etc. For example, when
the same person repeats the same observation, it is very likely that he may get different
readings everytime.
Least count error: The smallest value that can be measured by the measuring instrument
is called its least count. All the readings or measured values are good only
up to this value. The least count error is the error associated with the resolution of
the instrument. For example, a vernier callipers has the least count as 0.01 cm; a
spherometer may have a least count of 0.001 cm. Least count error belongs to the
category of random errors but within a limited size; it occurs with both systematic
and random errors. If we use a metre scale for measurement of length, it may have
graduations at 1 mm division scale spacing or interval.
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.
Absolute Error, Relative Error and Percentage Error
(a) Suppose the values obtained in several measurements are a1, a2, a3...., an. The
arithmetic mean of these values is taken as the best possible value of the quantity
under the given conditions of measurement as :
amean = (a1+a2+a3+...+an ) / n …………………… (4)
or, amean =
å=
n
t n ai/n ………………………………………… (5)
This is because, as explained earlier, it is reasonable to suppose that individual
measurements are as likely to overestimate as to underestimate the true value of
the quantity.
The magnitude of the difference between the true value of the quantity and
the individual measurement value is called the absolute error of the measurement.
This is denoted by | Δa | . In absence of any other method of knowing
true value, we considered arithmatic mean as the true value. Then the errors in the
individual measurement values are
Δa1 = amean – a1,
Δa2 = amean – a2,
.... .... ....
.... .... ....
Δan = amean – an
The Δa calculated above may be positive in certain cases and negative in some other
cases. But absolute error |Δa| will always be positive.
(b) 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 ……………………………….. (6)
=
å=
n
t n |Δat|/n …………………………………………………………………………………. (7)
If we do a single measurement, the value we get may be in the range
amean ± Δamean
i.e. a = amean ± Δamean
or,
amean – Δamean ≤ a ≤ amean + Δamean …………………………………………….. (8)
This implies that any measurement of the physical quantity a is likely to lie between
(amean+ Δamean) and (amean− Δamean).
(c) Instead of the absolute error, we often use the relative error or the percentage
error ( ¶ a). The relative error is the ratio of the mean absolute error Δamean to
the mean value amean of the quantity measured.
Relative error = Δamean/amean ……………………………………………………….. (9)
When the relative error is expressed in per cent, it is called the percentage error (
¶ a).
Thus, Percentage error
¶ a = (Δamean/amean) × 100% ………………………………………………………… (10)
Example: Two clocks are being tested against a standard clock located in a national
laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks
are :
Days Clock 1 Clock 2
Monday 12:00:05 10:15:06
Tuesday 12:01:15 10:14:59
Wednesday 11:59:08 10:15:18
Thursday 12:01:50 10:15:07
Friday 11:59:15 10:14:53
Saturday 12:01:30 10:15:24
Sunday 12:01:19 10:15:11
If you are doing an experiment that requires precision time interval measurements,
which of the two clocks will you prefer?
Answer: The range of variation over the seven days of observations is 162 s for
clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the
standard time than the average reading of clock 2. The important point is that a
clock’s zero error is not as significant for precision work as its variation, because a
‘zero-error’ can always be easily corrected. Hence clock 2 is to be preferred to clock
1.
Example: We measure the period of oscillation of a simple pendulum. In successive
measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s.
Calculate the absolute errors, relative error or percentage error.
Answer: The mean period of oscillation of the pendulum
2.624
5
T = 2.63 + 2.56 + 2.42 + 2.71+ 2.80 = s
As the periods are measured to a resolution of 0.01 s, all times are to the second
decimal; it is proper to put this mean period also to the second decimal.
The errors in the measurements are
2.63 s – 2.62 s = 0.01 s
2.56 s – 2.62 s = – 0.06 s
2.42 s – 2.62 s = – 0.20 s
2.71 s – 2.62 s = 0.09 s
2.80 s – 2.62 s = 0.18 s
Note that the errors have the same units as the quantity to be measured. The arithmetic
mean of all the absolute errors is
Δ Τmean = 0.11
5
0.01+ 0.06 + 0.20 + 0.09 + 0.18 =
s
That means, the period of oscillation of the simple pendulum is (2.62 ± 0.11) s i.e. it
lies between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s. As
the arithmetic mean of all the absolute errors is 0.11 s, there is already an error in
the tenth of a second. Hence there is no point in giving the period to a hundredth. A
more correct way will be to write
T = 2.6 ± 0.1 s
For this example, the relative error or the percentage error is
¶ a= 100 4%
2.6
0.1 ´ =
Combination of Errors
If we do an experiment involving several measurements, we must know how the errors
in all the measurements combine. For example, density is the ratio of the mass
to the volume of the substance. If we have errors in the measurement of mass and
of the sizes or dimensions, we must know what the error will be in the density of the
substance. To make such estimates, we should learn how errors combine in various
mathematical operations. For this, we use the following procedure.
(a) Error of a sum or 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. We wish to find the error ΔZ in
the sum
Z = A + B.
We have by addition, Z ± ΔZ
= (A ± ΔA) + (B ± ΔB).
The maximum possible error in Z
ΔZ = ΔA + ΔB
For the difference Z = A – B, we have
Z ± Δ Z = (A ± ΔA) – (B ± ΔB)
= (A – B) ± ΔA ± ΔB
or, ± ΔZ = ± ΔA ± ΔB
The maximum value of the error ΔZ is again
ΔA + ΔB.
Hence the 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.
Example: The temperatures of two bodies measured by a thermometer are
t1 = 20˚C ± 0.5˚C and t2 = 50˚C ± 0.5˚C. Calculate the temperature difference
and the error their in.
Answer t′ = t2–t1 = (50˚C±0.5˚C)– (20˚C±0.5˚C)
t′ = 30˚C ± 1˚C
(b) Error of a product or a quotient
Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then
Z ± ΔZ = (A ± ΔA) (B ± ΔB)
= AB ± B ΔA ± A ΔB ± ΔA ΔB.
Dividing LHS by Z and RHS by AB we have,
1±(ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).
Since ΔA and ΔB are small, we shall ignore their product. Hence the maximum relative
error
ΔZ/ Z = (ΔA/A) + (ΔB/B).
You can easily verify that this is true for division also.
Hence the rule: When two quantities are multiplied or divided, the relative
error in the result is the sum of the relative errors in the multipliers.
Example: The resistance R = V/I where V = (100 ± 5)V and I = (10 ± 0.2)A. Find
the percentage error in R.
Answer: The percentage error in V is 5% and in I it is 2%. The total error in R
would therefore be
5% + 2% = 7%.
Example: Two resistors of resistances R1 = 100 ±3 ohm and R2 = 200 ± 4 ohm are
connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a)
series combination, (b) parallel combination.
Use for
(a) the relation R =R1 + R2, and for
(b)
2
1
1
1
'
1
R R R
= +
and,
2²
2
1²
1
'²
'
R
R
R
R
R
D R = D + D
Answer:
(a) The equivalent resistance of series combination
R =R1 + R2 = (100 ± 3) ohm + (200 ± 4) ohm
= 300 ± 7 ohm.
(b) The equivalent resistance of parallel combination
200 3
1
100 3
1
'
1
±
+
±
=
R
200
= 3
66.66
3
Þ R'= 200 = ohm
Now,
2²
2
1²
1
'²
'
R
R
R
R
R
D R = D + D
=
200²
3
100²
3
66.7²
D R' = +
40000
Þ D R'= 66.7² ´ 12 + 3
=1.67; 1.7
Hence R’ = 66.7 ± 1.7
(c) Error in case of a measured quantity raised to a power
Suppose Z = A²,
Then, ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).
Hence, the relative error in A2 is two times the error in A.
In general, if Z = Ap Bq/Cr
Then, ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).
Hence the rule: The relative error in a physical quantity raised to the power
k is the k times the relative error in the individual quantity.
Example: Find the relative error in Z,
if Z = A4B1/3/CD3/2.
Answer The relative error in Z is
ΔZ/Z = 4(ΔA/A) +(1/3) (ΔB/B) + (ΔC/C) + (3/2) (ΔD/D).
Example: The period of oscillation of a simple pendulum is
g
T = 2p L
Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations
of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What
is the accuracy in the determination of g ?
Answer: As,
g
T = 2p L
So, g
T ² = 4p ² ´ L
Hence,
T²
g = 4p ² ´ L
Here,
n
T = t
And
n
D T = D t
So,
t
t
t
n
n
t
T
D T = D ´ = D
The errors in both L and t are the least count errors. Therefore,
(Δg/g) = (ΔL/L) + 2(ΔT/T )
= 0.032
90
2 1
20.0
0.1 = ÷ø
ö çè
+ æ
Thus, the percentage error in g is
100 (Δg/g) = 100(ΔL/L) + 2 ´ 100 (ΔT/T )
= 3%
SIGNIFICANT FIGURES
As discussed above, every measurement involves errors. Thus, the result of measurement
should be reported in a way that indicates the precision of measurement.
Normally, 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.
If we say the period of oscillation of a simple pendulum is 1.62 s, the digits 1 and 6
are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has
three significant figures. 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. Clearly, reporting the result of measurement that includes more digits
than the significant digits is superfluous and also misleading since it would give a
wrong idea about the precision of measurement.
The rules for determining the number of significant figures:
· A choice of change of different units does not change the number of
significant digits or figures in a measurement.
· All the non-zero digits are significant.
· All the zeros between two non-zero digits are significant, no matter
where the decimal point is, if at all.
· 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.
· The terminal or trailing zero(s) in a number without a decimal point
are not significant.
· The trailing zero(s) in a number with a decimal point are significant.
· To remove such ambiguities in determining the number of significant
figures, the best way is to report every measurement in scientific
notation (in the power of 10). In this notation, every number is expressed
as a × 10b, where a is a number between 1 and 10, and b is any positive or
negative exponent (or power) of 10.
· For a number greater than 1, without any decimal, the trailing zero(s)
are not significant.
· For a number with a decimal, the trailing zero(s) are significant.
Rules for Arithmetic Operations with Significant Figures:
· In multiplication or division, the final result should retain as many
significant figures as are there in the original number with the least
significant figures.
· In addition or subtraction, the final result should retain as many
decimal places as are there in the number with the least decimal
places.
Rounding off the Uncertain Digits:
· The preceding digit is raised by 1 if the insignificant digit to be
dropped (the underlined digit in this case) is more than 5, and is left
unchanged if the latter is less than 5. But what if the number is 2.745 in
which the insignificant digit is 5. Here, the convention is that if the preceding
digit is even, the insignificant digit is simply dropped and, if it is
odd, the preceding digit is raised by 1.
Example: Each side of a cube is measured to be 7.203 m. What are the total surface
area and the volume of the cube to appropriate significant figures?
Answer: The number of significant figures in the measured length is 4. The calculated
area and the volume should therefore be rounded off to 4 significant figures.
Surface area of the cube = 6(7.203)² m²
= 311.299254 m²
= 311.3 m²
Volume of the cube = (7.203)³ m³
= 373.714754 m³
= 373.7 m³
Example: 5.74 g of a substance occupies 1.2 cm³. Express its density by keeping
the significant figures in view.
Answer: There are 3 significant figures in the measured mass whereas there are
only 2 significant figures in the measured volume. Hence the density should be expressed
to only 2 significant figures.
Density = 4.8
1.2
5.74 = g/cm³
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Finish Line & Beyond
Rules for Determining the Uncertainty in the Results of Arithmatic Calculations:
(1) If the length and breadth of a thin rectangular sheet are measured, using a
metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures
in each measurement. It means that the length l may be written as
l = 16.2 ± 0.1 cm
= 16.2 cm ± 0.6 %.
Similarly, the breadth b may be written as
b = 10.1 ± 0.1 cm
= 10.1 cm ± 1 %
Then, the error of the product of two (or more)
experimental values, using the combination of
errors rule, will be
lb = 163.62 cm2 + 1.6%
= 163.62 + 2.6 cm2
This leads us to quote the final result as
lb = 164 + 3 cm2
Here 3 cm2 is the uncertainty or error in the estimation of area of rectangular sheet.
(2) If a set of experimental data is specified to n significant figures, a result
obtained by combining the data will also be valid to n significant figures.
However, if data are subtracted, the number of significant figures can be reduced.
(3) The relative error of a value of number specified to significant figures
depends not only on n but also on the number itself.
For example, the accuracy in measurement of mass 1.02 g is ± 0.01 g whereas another
measurement 9.89 g is also accurate to ± 0.01 g.
The relative error in 1.02 g is
= (± 0.01/1.02) ´ 100 %
= ± 1%
Similarly, the relative error in 9.89 g is
= (± 0.01/9.89) ´ 100 %
= ± 0.1 %
Finally, remember that intermediate results in a multi-step computation
should be calculated to one more significant figure in every measurement
than the number of digits in the least precise measurement.
DIMENSIONS OF PHYSICAL QUANTITIES
The nature of a physical quantity is described by its dimensions. All the physical
quantities represented by derived units can be expressed in terms of some combination
of seven fundamental or base quantities. We shall call these base quantities as
the seven dimensions of the physical world, which are denoted with square brackets
[ ]. Thus, length has the dimension [L], mass [M], time [T], electric current [A],
thermodynamic temperature [K], luminous intensity [cd], and amount of substance
[mol].
The dimensions of a physical quantity are the powers (or exponents) to
which the base quantities are raised to represent that quantity.
Note that using the square brackets [ ] round a quantity means that we are dealing
with ‘the dimensions of’ the quantity. In mechanics, all the physical quantities can
be written in terms of the dimensions [L], [M] and [T]. For example, the volume occupied
by an object is expressed as the product of length, breadth and height, or
three lengths.
Hence the dimensions of volume are
[L] × [L] × [L] = [L]³ = [L³].
As the volume is independent of mass and time, it is said to possess zero dimension
in mass [M°], zero dimension in time [T°] and three dimensions in length.
Similarly, force, as the product of mass and acceleration, can be expressed as
Force = mass × acceleration
= mass × (length)/(time)²
The dimensions of force are
[M] [L]/[T]² =[M L T–2].
Thus, the force has one dimension in mass, one dimension in length, and –2 dimensions
in time. The dimensions in all other base quantities are zero.
Note that in this type of representation, the magnitudes are not considered. It is the
quality of the type of the physical quantity that enters. Thus, a change in velocity,
initial velocity, average velocity, final velocity, and speed are all equivalent in this
context. Since all these quantities can be expressed as length/time, their dimensions
are [L]/[T] or [L T–1].
Dimensional Formulae and Dimensional Equations
Dimensional Formula: 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. For example, the dimensional formula of the
volume is [M° L3 T°], and that of speed or velocity is [M° L T-1]. Similarly, [M° L T–2]
is the dimensional formula of acceleration and [M L–3 T°] that of mass density.
Dimensional Equation: An equation obtained by equating a physical quantity with
its dimensional formula is called the dimensional equation of the physical quantity.
Thus, the dimensional equations are the equations, which represent the dimensions
of a physical quantity in terms of the base quantities.
For example, the dimensional equations of volume [V], speed [v], force [F ] and
mass density [ ρ] may be expressed as
[V] = [M0 L3 T0]
[v] = [M0 L T–1]
[F] = [M L T–2]
[ ρ] = [M L–3 T0]
Dimensional Analysis and Its Applications:
The recognition of concepts of dimensions, which guide the description of physical
behaviour is of basic importance as only those physical quantities can be added or
subtracted which have the same dimensions. A thorough understanding of dimensional
analysis helps us in deducing certain relations among different physical quantities
and checking the derivation, accuracy and dimensional consistency or homogeneity
of various mathematical expressions. 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. The same is true for dimensions of a physical quantity. Similarly, physical
quantities represented by symbols on both sides of a mathematical equation
must have the same dimensions.
Checking the Dimensional Consistency of Equations:
The magnitudes of physical quantities may be added together or subtracted from one
another only if they have the same dimensions. In other words, we can add or subtract
similar physical quantities. Thus, velocity cannot be added to force, or an electric
current cannot be subtracted from the thermodynamic temperature. This simple
principle called the principle of homogeneity of dimensions in an equation is extremely
useful in checking the correctness of an equation. If the dimensions of all the
terms are not same, the equation is wrong.
Hence, if we derive an expression for the length (or distance) of an object, regardless
of the symbols appearing in the original mathematical relation, when all the individual
dimensions are simplified, the remaining dimension must be that of length.
Similarly, if we derive an equation of speed, the dimensions on both the sides of
equation, when simplified, must be of length/ time, or [L T–1].
Dimensions are customarily used as a preliminary test of the consistency of an equation,
when there is some doubt about the correctness of the equation. However, the
dimensional consistency does not guarantee correct equations. It is uncertain to the
extent of dimensionless quantities or functions. The arguments of special functions,
such as the trigonometric, logarithmic and exponential functions must be dimensionless.
A pure number, ratio of similar physical quantities, such as angle as the ratio
(length/length), refractive index as the ratio (speed of light in vacuum/speed of light
in medium) etc., has no dimensions.
Example: Let us consider an equation
mv2 = mgh
2
1
where m is the mass of the body, v its velocity, g is the acceleration due to gravity
and h is the height. Check whether this equation is dimensionally correct.
Answer: The dimensions of LHS are
[M] [L T–1 ]2 = [M] [ L2 T–2]
= [M L2 T–2]
The dimensions of RHS are
[M][L T–2] [L] = [M][L2 T–2]
= [M L2 T–2]
The dimensions of LHS and RHS are the same and hence the equation is dimensionally
correct.
Example: The SI unit of energy is J = kg m2 s–2; that of speed v is m s–1 and of acceleration
a is m s–2. Which of the formulae for kinetic energy (K) given below
can you rule out on the basis of dimensional arguments (m stands for the mass of
the body) :
(a) K = m2v3
(b) K = (1/2)mv2
(c) K = ma
(d) K = (3/16)mv2
(e) K = (1/2)mv2 + ma
Answer: Every correct formula or equation must have the same dimensions on both
sides of the equation. Also, only quantities with the same physical dimensions can be
added or subtracted. The dimensions of the quantity on the right side are
(a) [M2 L3 T–3]
(b) & (d) [M L2 T–2]
(c) [M L T–2].
(e) No proper dimension since two quantities of different dimensions have been added.
Since the kinetic energy K has the dimensions of [M L2 T–2], formulas (a), (c) and (e)
are ruled out.
Note that dimensional arguments cannot tell which of the two, (b) or (d), is the correct
formula. For this, one must turn to the actual definition of kinetic energy. The
correct formula for kinetic energy is given by (b).
Deducing Relation among the Physical Quantities:
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. Let us take an example.
Example: 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.
Answer: The dependence of time period T on the quantities l, g and m as a product
may be written as:
T = kl x g ymz
where k is dimensionless constant and x, y and z are the exponents.
By considering dimensions on both sides, we have
[L0M0T1 ] = [L1][L1T-2]y[M]z
= Lx+yT–2yMz
On equating the dimensions on both sides, we have
x + y = 0; –2y = 1; and z = 0
So that
2
x = 1 ,
2
y = - 1 and z=0
Then, T = k l ½ g–½
