08
Units and
Measurements
Quick Revision
1. Physical Quantities All the quantities which ● Derived Units The units of measurement
can be measured directly or indirectly and of all other physical quantities, which can
in terms of which laws of Physics are described be obtained from fundamental units are
are called physical quantities. called derived units.
These can be divided into two types, namely 3. System of Units It is the complete set of
fundamental and derived quantities. units, both fundamental and derived physical
● Fundamental Quantities The physical units.
quantities which are independent of other The common system of units used in
physical quantities and are not defined in mechanics are as follows
terms of other physical quantities are called ● The FPS System It is the British
fundamental or base quantities. engineering system of units. It uses foot as
e.g. Mass, length, time, temperature, luminous the unit of length, pound as the unit of mass
intensity, electric current, amount of and second as the unit of time.
substance, etc. ● The CGS System It is the French system of
● Derived Quantities Those quantities which
units, which uses centimetre, gram and
can be derived from the fundamental physical second as the units of length, mass and time,
quantities are called derived quantities. respectively.
e.g. Velocity, acceleration, linear momentum, ● The MKS System It uses metre, kilogram
etc. and second as the fundamental units of
2. Physical Unit The standard amount of a length, mass and time, respectively.
physical quantity chosen to measure the ● The International System of Units
physical quantity of same kind is called a (SI Units) The system of units which is
physical unit. The physical units can be accepted internationally for measurement is
classified into following two types the ‘Systeme International d’ Units (French
● Fundamental Units The units of for International System of Units),
fundamental quantities are known as abbreviated as SI.
fundamental units.
,4. Fundamental Quantities and their SI Units For Area
-28
Fundamental SI unit Symbol
● Barn, 1 barn = 10 m2
● Acre, 1 acre = 4047 m
2
quantity
4
● Hectare, 1 hectare = 10 m2
Length metre m
7. Accuracy and Precision of Instruments
Mass kilogram kg
The accuracy of a measurement is a measure of
Time second s how close the measured value is to the true
Electric current ampere A value of the quantity. While, precision tells us
to what resolution or limit the quantity is
Temperature kelvin K measured.
Amount of mole mol 8. Errors in Measurement
substance
Difference in the true value and the measured
Luminous intensity candela cd value of a quantity is called error in
measurement.
5. Supplementary Quantities and their SI Units
Error = True value – Measured value
Supplementary SI unit Symbol In general, the errors can be further classified as
quantity ● Systematic Errors Those errors that tend
Plane angle radian rad to be in one direction, either positive or
Solid angle steradian sr negative are called systematic errors. Some
of the sources of systematic errors are
6. Some Important Practical Units (a) Instrumental errors They occur due to
For Length/Distance imperfect design or manufacture or
● Astronomical Unit It is the mean distance calibration of the measuring instrument.
of the earth from the sun. (b) Imperfection in experimental
1 AU = 1.496 ´ 10 11 m technique or procedure These types of
errors occur due to the experimental
● Light year It is the distance travelled by
arrangement limitations.
light in vacuum in one year.
(c) Personal errors These errors arise due
1 ly = 9.46 ´ 10 15 m
-6
to inexperience of the observer such as lack
● Micro or micrometer, 1 mm = 10 m of proper setting of the apparatus and
● Nanometer, 1 nm = 10 -9 m taking observations without observing
proper precautions, etc.
● Angstrom, 1 Å = 10 -10 m
(d) Errors due to external causes Various
● Fermi This unit is used for measuring parameters such as change in
nuclear sizes. 1 Fm = 10 -15 m temperature, pressure, volume, etc. during
For Mass experiment may affect the reading of
● Pound, 1 lb = 0.4536 kg
measurement.
● Random Errors The errors which occur
● Slug, 1 slug = 14.59 kg
irregularly and are random in magnitude
● Quintal, 1 q = 100 kg and direction are called random errors.
● Tonne or metric tonne, 1 t = 1000 kg ● Least Count Error The smallest value that
can be measured by a measuring instrument
● Atomic mass unit (It is defined as the is called the least count of the instrument
(1/12)th of the mass of one 12
6
C-atom) 1 u or and error in its value is called least count
amu = 1.66 ´ 10 –27 kg. error.
, ● Absolute Error The magnitude of the ● Error in Case of a Measured Quantity
difference between the true value of the Raised to a Power
quantity and the individual measured value Relative error of Z = A n B m is
is called the absolute error of the
measurement. It is denoted by | D a |. DZ é DA DB ù
= ± ên +m
Suppose, the measured values are Z ë A B úû
a 1, a 2, a 3, K , a n , then arithmetic mean of 10. Significant Figures
a + a2 + K + an
these values is a Mean = 1 . The digits that are known reliably plus the first
n uncertain digit are known as significant digits
If we take arithmetic mean a Mean as the true or significant figures.
value, then the absolute errors in the Rules for Determining the Number of
individual measured value will be Significant Figures
Da 1 = a Mean – a 1
Rule 1 All non-zero digits are significant.
● Mean Absolute Error It is the arithmetic
mean of the magnitudes of absolute errors in Rule 2 All the zeros between two non-zero
all the measurements of the quantity. digits are significant, no matter where
n | Da |
the decimal point is, if at all.
Mean absolute error, D a Mean = S i
Rule 3 If the number is less than one, the
i =1 n
zero(s) on the right of decimal point
● Relative Error or Fractional Error It is and to the left of first non-zero digit
defined as the ratio of mean absolute error to are not significant.
the mean value of the quantity measured.
Rule 4 In a number without a decimal point,
D a Mean
Relative error, da = the terminal or trailing zeros is not
a Mean significant.
● Percentage Error When fractional error or Rule 5 The trailing zero(s) in a number with a
relative error is expressed in percentage, decimal points are significant.
then it is called percentage error. 11. Rules for Arithmetical Operations with
D a Mean Significant Figures
Percentage error, da % = ´ 100%
a Mean Some rules of arithmetical operations with
9. Combination of Errors significant figures are as given below
● Addition and Subtraction In both,
● Error in Sum or Difference
addition and subtraction, the final result
Let X = A + B or X = A - B
should retain as many decimal places as
where, A and B are physical quantities have are there in the number with the least
measured value A ± DA, B ± DB , decimal places.
respectively. ● Multiplication and Division In
So, the maximum possible error in sum and multiplication or division, the final result
difference, DZ = DA + DB should retain as many significant figures, as
● Error in Product or Quotient are there in the original number with the
least significant figures.
A
Let X = A ´ B or X = 12. Rounding Off The result of computation with
B
approximate numbers, which contains more
So, the maximum possible error in product than one uncertain digit, should be rounded
DZ DA DB off. While rounding off measurements, we use
or quotient is = +
Z A B the following rules by convention
Units and
Measurements
Quick Revision
1. Physical Quantities All the quantities which ● Derived Units The units of measurement
can be measured directly or indirectly and of all other physical quantities, which can
in terms of which laws of Physics are described be obtained from fundamental units are
are called physical quantities. called derived units.
These can be divided into two types, namely 3. System of Units It is the complete set of
fundamental and derived quantities. units, both fundamental and derived physical
● Fundamental Quantities The physical units.
quantities which are independent of other The common system of units used in
physical quantities and are not defined in mechanics are as follows
terms of other physical quantities are called ● The FPS System It is the British
fundamental or base quantities. engineering system of units. It uses foot as
e.g. Mass, length, time, temperature, luminous the unit of length, pound as the unit of mass
intensity, electric current, amount of and second as the unit of time.
substance, etc. ● The CGS System It is the French system of
● Derived Quantities Those quantities which
units, which uses centimetre, gram and
can be derived from the fundamental physical second as the units of length, mass and time,
quantities are called derived quantities. respectively.
e.g. Velocity, acceleration, linear momentum, ● The MKS System It uses metre, kilogram
etc. and second as the fundamental units of
2. Physical Unit The standard amount of a length, mass and time, respectively.
physical quantity chosen to measure the ● The International System of Units
physical quantity of same kind is called a (SI Units) The system of units which is
physical unit. The physical units can be accepted internationally for measurement is
classified into following two types the ‘Systeme International d’ Units (French
● Fundamental Units The units of for International System of Units),
fundamental quantities are known as abbreviated as SI.
fundamental units.
,4. Fundamental Quantities and their SI Units For Area
-28
Fundamental SI unit Symbol
● Barn, 1 barn = 10 m2
● Acre, 1 acre = 4047 m
2
quantity
4
● Hectare, 1 hectare = 10 m2
Length metre m
7. Accuracy and Precision of Instruments
Mass kilogram kg
The accuracy of a measurement is a measure of
Time second s how close the measured value is to the true
Electric current ampere A value of the quantity. While, precision tells us
to what resolution or limit the quantity is
Temperature kelvin K measured.
Amount of mole mol 8. Errors in Measurement
substance
Difference in the true value and the measured
Luminous intensity candela cd value of a quantity is called error in
measurement.
5. Supplementary Quantities and their SI Units
Error = True value – Measured value
Supplementary SI unit Symbol In general, the errors can be further classified as
quantity ● Systematic Errors Those errors that tend
Plane angle radian rad to be in one direction, either positive or
Solid angle steradian sr negative are called systematic errors. Some
of the sources of systematic errors are
6. Some Important Practical Units (a) Instrumental errors They occur due to
For Length/Distance imperfect design or manufacture or
● Astronomical Unit It is the mean distance calibration of the measuring instrument.
of the earth from the sun. (b) Imperfection in experimental
1 AU = 1.496 ´ 10 11 m technique or procedure These types of
errors occur due to the experimental
● Light year It is the distance travelled by
arrangement limitations.
light in vacuum in one year.
(c) Personal errors These errors arise due
1 ly = 9.46 ´ 10 15 m
-6
to inexperience of the observer such as lack
● Micro or micrometer, 1 mm = 10 m of proper setting of the apparatus and
● Nanometer, 1 nm = 10 -9 m taking observations without observing
proper precautions, etc.
● Angstrom, 1 Å = 10 -10 m
(d) Errors due to external causes Various
● Fermi This unit is used for measuring parameters such as change in
nuclear sizes. 1 Fm = 10 -15 m temperature, pressure, volume, etc. during
For Mass experiment may affect the reading of
● Pound, 1 lb = 0.4536 kg
measurement.
● Random Errors The errors which occur
● Slug, 1 slug = 14.59 kg
irregularly and are random in magnitude
● Quintal, 1 q = 100 kg and direction are called random errors.
● Tonne or metric tonne, 1 t = 1000 kg ● Least Count Error The smallest value that
can be measured by a measuring instrument
● Atomic mass unit (It is defined as the is called the least count of the instrument
(1/12)th of the mass of one 12
6
C-atom) 1 u or and error in its value is called least count
amu = 1.66 ´ 10 –27 kg. error.
, ● Absolute Error The magnitude of the ● Error in Case of a Measured Quantity
difference between the true value of the Raised to a Power
quantity and the individual measured value Relative error of Z = A n B m is
is called the absolute error of the
measurement. It is denoted by | D a |. DZ é DA DB ù
= ± ên +m
Suppose, the measured values are Z ë A B úû
a 1, a 2, a 3, K , a n , then arithmetic mean of 10. Significant Figures
a + a2 + K + an
these values is a Mean = 1 . The digits that are known reliably plus the first
n uncertain digit are known as significant digits
If we take arithmetic mean a Mean as the true or significant figures.
value, then the absolute errors in the Rules for Determining the Number of
individual measured value will be Significant Figures
Da 1 = a Mean – a 1
Rule 1 All non-zero digits are significant.
● Mean Absolute Error It is the arithmetic
mean of the magnitudes of absolute errors in Rule 2 All the zeros between two non-zero
all the measurements of the quantity. digits are significant, no matter where
n | Da |
the decimal point is, if at all.
Mean absolute error, D a Mean = S i
Rule 3 If the number is less than one, the
i =1 n
zero(s) on the right of decimal point
● Relative Error or Fractional Error It is and to the left of first non-zero digit
defined as the ratio of mean absolute error to are not significant.
the mean value of the quantity measured.
Rule 4 In a number without a decimal point,
D a Mean
Relative error, da = the terminal or trailing zeros is not
a Mean significant.
● Percentage Error When fractional error or Rule 5 The trailing zero(s) in a number with a
relative error is expressed in percentage, decimal points are significant.
then it is called percentage error. 11. Rules for Arithmetical Operations with
D a Mean Significant Figures
Percentage error, da % = ´ 100%
a Mean Some rules of arithmetical operations with
9. Combination of Errors significant figures are as given below
● Addition and Subtraction In both,
● Error in Sum or Difference
addition and subtraction, the final result
Let X = A + B or X = A - B
should retain as many decimal places as
where, A and B are physical quantities have are there in the number with the least
measured value A ± DA, B ± DB , decimal places.
respectively. ● Multiplication and Division In
So, the maximum possible error in sum and multiplication or division, the final result
difference, DZ = DA + DB should retain as many significant figures, as
● Error in Product or Quotient are there in the original number with the
least significant figures.
A
Let X = A ´ B or X = 12. Rounding Off The result of computation with
B
approximate numbers, which contains more
So, the maximum possible error in product than one uncertain digit, should be rounded
DZ DA DB off. While rounding off measurements, we use
or quotient is = +
Z A B the following rules by convention
