Chapter 2 - Units and Measurement
Chapter 2 - Units and Measurement
Chapter 2 - Units and Measurement
COM } ON GOOGE
Mass kilogram kg
Time second s
Thermo dynamic
Temperature kelvin K
Amount of
Substance mole mol
Grapes Time
10-3 Interval
Event (s)
Human
102 Life span of most unstable particle
10-24
Automobile
103 Period of x-rays
10-19
Boeing 747 aircraft
108 Period of light wave
10-15
Moon
1023 Period of radio wave
10-6
Earth
1025 Period of sound wave
10-3
Sun
1030 Wink on an eye
10-1
Milky way Galaxy
1041 Travel time of light from moon to
earth
Observable Universe 100
1055
2. Imperfections in experimental techniques: If
Travel time of light from sun to earth
102 the technique is not accurate (for example
measuring temperature of human body by
placing thermometer under armpit resulting in
Rotation period of the earth
105 lower temperature than actual) and due to the
external conditions like temperature, wind,
Revolution period of the earth humidity, these kinds of errors occur.
107 3. Personal errors: Errors occurring due to human
carelessness, lack of proper setting, taking down
Average human life span incorrect reading are called personal errors.
109
These errors can be removed by:
Age of Egyptian pyramids o Taking proper instrument and calibrating
1011 it properly.
o Experimenting under proper atmospheric
Time since dinosaur extinction
1015 conditions and techniques.
Removing human bias as far as possible
Age of Universe
1017 Random Errors
Errors which occur at random with respect to sign
and size are called Random errors.
Accuracy and Precision of Instruments • These occur due to unpredictable fluctuations
• Any uncertainty resulting from measurement in experimental conditions like temperature,
by a measuring instrument is called an error. voltage supply, mechanical vibrations, personal
They can be systematic or random. errors etc.
• Accuracy of a measurement is how close the Least Count Error
measured value is to the true value. Smallest value that can be measured by the
• Precision is the resolution or closeness of a measuring instrument is called its least
series of measurements of a same quantity count. Least count error is the error associated
under similar conditions. with the resolution or the least count of the
• If the true value of a certain length is 3.678 cm instrument.
and two instruments with different • Least count errors can be minimized by using
resolutions, up to 1 (less precise) and 2 (more instruments of higher precision/resolution and
precise) decimal places respectively, are used. improving experimental techniques (taking
If first measures the length as 3.5 and the several readings of a measurement and then
second as 3.38 then the first has more taking a mean).
accuracy but less precision while the second
Errors in a series of Measurements
has less accuracy and more precision.
Suppose the values obtained in several
Types of Errors- Systematic Errors measurement are a1, a2, a3, …, an.
Errors which can either be positive or negative are Arithmetic mean, amean = (a1+ a2 + a3+ … + an)/n
called Systematic errors. They are of following 𝑛
𝑎𝑖
types: 𝑎𝑚𝑒𝑎𝑛 = ∑
1. Instrumental errors: These arise from imperfect 𝑛
𝑖=1
design or calibration error in the instrument. • Absolute Error: The magnitude of the
Worn off scale, zero error in a weighing scale are difference between the true value of the
some examples of instrument errors. quantity and the individual measurement value
is called absolute error of the measurement. It
is denoted by |Δa| (or Mod of Delta a). The
Raised
mod value is always positive even if Δa is
Sum or to
negative. The individual errors are:
Criteria Difference Product Power
Δa1 = amean - a1
Δa2 = amean - a2,
… … … Resultant
… … … value Z Z=A±B Z = AB
Z = Ak
Δan = amean – an
• Mean absolute error is the arithmetic mean of
Z ± ΔZ = (A Z ± ΔZ =
all absolute errors. It is represented by Δamean. Z ± ΔZ =
|𝛥𝑎1| + |𝛥𝑎2| + |𝛥𝑎3| + … . +|𝛥𝑎𝑛| Result ± ΔA) + (B (A ± ΔA)
𝛥𝑎𝑚𝑒𝑎𝑛 = with error ± ΔB) (B ± ΔB) (A ±
𝑛 ΔA)k
𝑛
|∆𝑎𝑖 | Resultant ΔZ/Z =
𝛥𝑎𝑚𝑒𝑎𝑛 = ∑
𝑛 error ± ΔZ = ± ΔA/A ±
𝑖=1
range ΔA ± ΔB ΔB/B
For single measurement, the value of ‘a’ is
always in the range 𝑎𝑚𝑒𝑎𝑛 ± 𝛥𝑎𝑚𝑒𝑎𝑛
So, 𝑎 = 𝑎𝑚𝑒𝑎𝑛 ± 𝛥𝑎𝑚𝑒𝑎𝑛 ΔZ/Z =
Or, 𝑎𝑚𝑒𝑎𝑛 − 𝛥𝑎𝑚𝑒𝑎𝑛 ≤ 𝑎 ≤ 𝑎𝑚𝑒𝑎𝑛 + 𝛥𝑎𝑚𝑒𝑎𝑛 Maximum ΔZ = ΔA + ΔA/A + ΔZ/Z =
error ΔB ΔB/B k(ΔA/A)
• Relative Error: It is the ratio of mean absolute
error to the mean value of the quantity
measured. Sum of Sum of k times
𝛥𝑎𝑚𝑒𝑎𝑛 absolute relative relative
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
𝑎𝑚𝑒𝑎𝑛 Error errors errors error
• Percentage Error: It is the relative error
expressed in percentage. It is denoted by δa.
𝛥𝑎𝑚𝑒𝑎𝑛
𝛿𝑎 = × 100%
𝑎𝑚𝑒𝑎𝑛 Significant Figures
Combinations of Errors Every measurement results in a number that
If a quantity depends on two or more other includes reliable digits and uncertain digits.
quantities, the combination of errors in the two Reliable digits plus the first uncertain digit are
quantities helps to determine and predict the called significant digits or significant figures.These
errors in the resultant quantity. There are several indicate the precision of measurement which
procedures for this. depends on least count of measuring instrument.
Example, period of oscillation of a pendulum is 1.62
Suppose two quantities A and B have values as A ± s. Here 1 and 6 are reliable and 2 is uncertain. Thus,
ΔA and B ± ΔB. Z is the result and ΔZ is the error the measured value has three significant figures.
due to combination of A and B.
Rules for determining number of significant figures
• All non-zero digits are significant.
• All zeros between two non-zero digits are
significant irrespective of decimal place.
• For a value less than 1, zeroes after decimal and
before non-zero digits are not significant. Zero
before decimal place in such a number is always
insignificant.
• Trailing zeroes in a number without decimal
if mass = 4.237 g (4 227.2 (1 digit
place are insignificant.
significant figures) after decimal)
• Trailing zeroes in a number with decimal place and Volume = 2.51 & .301 (3
are significant. cm3(3 significant digits after
Cautions to remove ambiguities in determining figures) decimal) is
number of significant figures
• Change of units should not change number of = 663.821
significant digits. Example, 4.700m = 470.0 cm Density = 4.237
= 4700 mm. In this, first two quantities have 4 g/2.51 cm3 =
but third quantity has 2 significant figures. 1.68804 g cm-3 =
1.69 g cm-3 (3 Since 227.2 is
• Use scientific notation to report
significant figures) precise up to
measurements. Numbers should be expressed
only 1
in powers of 10 like a x 10b where b is
decimal
called order of magnitude. Example,
place, Hence,
4.700 𝑚 = 4.700 × 102 𝑐𝑚 = 4.700 ×
the final
103 𝑚𝑚 = 4.700 × 10−3 𝑘𝑚
result should
In all the above, since power of 10 are
be 663.8
irrelevant, number of significant figures are 4.
• Multiplying or dividing exact numbers can have
infinite number of significant digits. Example,
radius = diameter / 2. Here 2 can be written as
Rules for Rounding off the uncertain digits
2, 2.0, 2.00, 2.000 and so on.
Rounding off is necessary to reduce the number of
insignificant figures to adhere to the rules of
Rules for Arithmetic operation with Significant arithmetic operation with significant figures.
Figures
Example
Multiplication or Addition or
(roundin
Type Division Subtraction
g off to
two
The final Rule Insignifica Preceding decimal
result should Number nt Digit Digit places)
retain as
The final result many decimal
Insignifica
should retain as places as nt digit to Number
many significant there in be Preceding – 3.137
figures as there in the original
dropped digit is
the original number with
is more raised by Result –
number with the the least
1 than 5 1. 3.14
lowest number of decimal
Rule significant digits. places.
Insignifica
Addition of nt digit to Number
Density = Mass / be Preceding
436.32 (2 – 3.132
Volume dropped digit is left
digits after
is less unchange Result –
decimal), 2 than 5 d. 3.13
Example
Example 12.9 - 7.06 = 5.84 or 5.8 (rounding off to
If
lowest number of decimal places of original
Insignifica preceding
number).
nt digit to digit is Number
be even, it is – 3.125 2. The relative error of a value of number specified
dropped left to significant figures depends not only on n but
is equal to unchange Result – also on the number itself.
3 5 d. 3.12 Example, accuracy for two numbers 1.02 and 9.89
is ±0.01. But relative errors will be: