Units and Measurements
SHORT NOTES
Fundamental Quantity Derived Quantity Significant Figure or Digits
The physical quantities which Those quantities which can Rules to find out the number of significant figures:
do not depend on any other be expressed in terms of 1. Rule: All the non-zero digits are significant e.g., 1984 has
physical quantities for their fundamental/base quantities. 4 SF.
measurements. e.g., Angle, speed or velocity 2. Rule: All the zeros between two non-zero digits are significant.
e.g., Mass, Length, Time Acceleration, force etc. e.g., 10806 has 5 SF.
Temperature, current, luminous 3. Rule: All the zeros to the left of first non-zero digit are not
Intensity & mole significant. e.g., 00108 has 3 SF.
4. Rule: If the number is less than 1, zeros on the right of the
System of Units decimal point but to the left of the first non-zero digit are not
(a) FPS System: Here length is measured in foot, mass in significant. e.g., 0.002308 has 4 SF.
pounds and time in second. 5. Rule: The trailing zeros (zeros to the right of the last non-zero
digit) in a number with a decimal point are significant. e.g.,
(b) CGS System: In this system, L is measured in cm, M is
01.080 has 4 SF.
measured in g and T is measured in sec.
6. Rule: The trailing zeros in a number without a decimal point
(c) MKS System: In this system, L is measured in metre, M is are not significant e.g., 010100 has 3 SF. But if the number
measured in kg and T is measured in sec. comes from some actual measurement then the trailing zeros
become significant. e.g., m = 100 kg has 3 SF.
Principle of Homogeneity
7. Rule: When the number is expressed in exponential form,
According to this, the physical quantities having same dimension the exponential term does not affect the number of S.F. For
can be added or subtracted with each other and for a given equation, example in x = 12.3 = 1.23 × 101 = .123 × 102 = 0.0123 × 103
dimensions of both sides must be same. = 123 × 10–1, each term has 3 SF only.
B Rules for arithmetical operations with significant figures:
For eg, in equation F
= A m+ +C ,
v 1. Rule: In addition or subtraction the number of decimal places
all the three parts of R.H.S have same dimension as force on L.H.S. in the result should be equal to the number of decimal places
of that term in the operation which contain lesser number of
Dimensions decimal places. e.g., 12.587 – 12.5 = 0.087 = 0.1 ( second
term contain lesser i.e., one decimal place)
The fundamental or base quantities along with their powers needed
to express a physical quantity is called dimensions 2. Rule: In multiplication or division, the number of SF in the
product or quotient is same as the smallest number of SF in
e.g., [F] = [MLT–2] is dimension of force.
any of the factors. e.g., 5.0 × 0.125 = 0.625 = 0.62.
Usage of Dimensional Analysis
Rounding Off
(i) To check the correctness of a given formula.
Rules for rounding off the numbers:
(ii) To establish relation between quantities dimensionally.
1. Rule: If the digit to be rounded off is more than 5, then the
(iii) To convert the value of a quantity from one system of
preceding digit is increased by one. e.g., 6.87≈ 6.9
units to other system.
2. Rule: If the digit to be rounded off is less than 5, than the
Limitations of Dimensional Analysis preceding digit is unaffected and is left unchanged. e.g., 3.94
(i) It does not predict the numerical value or number ≈ 3.9
associated with a physical quantity in a relation 3. Rule: If the digit to be rounded off is 5 then the preceding
digit is increased by one if it is odd and is left unchanged if it
e.g., v= u + 1 at & v = u + at
is even. e.g., 14.35 ≈ 14.4 and 14.45 ≈ 14.4
3 5
Both are dimensionally valid. Representation of Errors
(ii) It does not derive any relations involving trigonometric, 1. Mean absolute error is defined as
logarithmic and exponential functions ∆a1 + ∆a2 + ... + ∆an n
∆a
e.g., P = P0e–bt cannot be derived dimensionally.=
2 ∆a = ∑ i
n i =1 n
(iii) It does not give any information about dimensionally
Final result of measurement may be written as:
constants or nature of a quantity (vector/scalar) associated
with a relation. a = am ± ∆a
1
SHORT NOTES
Fundamental Quantity Derived Quantity Significant Figure or Digits
The physical quantities which Those quantities which can Rules to find out the number of significant figures:
do not depend on any other be expressed in terms of 1. Rule: All the non-zero digits are significant e.g., 1984 has
physical quantities for their fundamental/base quantities. 4 SF.
measurements. e.g., Angle, speed or velocity 2. Rule: All the zeros between two non-zero digits are significant.
e.g., Mass, Length, Time Acceleration, force etc. e.g., 10806 has 5 SF.
Temperature, current, luminous 3. Rule: All the zeros to the left of first non-zero digit are not
Intensity & mole significant. e.g., 00108 has 3 SF.
4. Rule: If the number is less than 1, zeros on the right of the
System of Units decimal point but to the left of the first non-zero digit are not
(a) FPS System: Here length is measured in foot, mass in significant. e.g., 0.002308 has 4 SF.
pounds and time in second. 5. Rule: The trailing zeros (zeros to the right of the last non-zero
digit) in a number with a decimal point are significant. e.g.,
(b) CGS System: In this system, L is measured in cm, M is
01.080 has 4 SF.
measured in g and T is measured in sec.
6. Rule: The trailing zeros in a number without a decimal point
(c) MKS System: In this system, L is measured in metre, M is are not significant e.g., 010100 has 3 SF. But if the number
measured in kg and T is measured in sec. comes from some actual measurement then the trailing zeros
become significant. e.g., m = 100 kg has 3 SF.
Principle of Homogeneity
7. Rule: When the number is expressed in exponential form,
According to this, the physical quantities having same dimension the exponential term does not affect the number of S.F. For
can be added or subtracted with each other and for a given equation, example in x = 12.3 = 1.23 × 101 = .123 × 102 = 0.0123 × 103
dimensions of both sides must be same. = 123 × 10–1, each term has 3 SF only.
B Rules for arithmetical operations with significant figures:
For eg, in equation F
= A m+ +C ,
v 1. Rule: In addition or subtraction the number of decimal places
all the three parts of R.H.S have same dimension as force on L.H.S. in the result should be equal to the number of decimal places
of that term in the operation which contain lesser number of
Dimensions decimal places. e.g., 12.587 – 12.5 = 0.087 = 0.1 ( second
term contain lesser i.e., one decimal place)
The fundamental or base quantities along with their powers needed
to express a physical quantity is called dimensions 2. Rule: In multiplication or division, the number of SF in the
product or quotient is same as the smallest number of SF in
e.g., [F] = [MLT–2] is dimension of force.
any of the factors. e.g., 5.0 × 0.125 = 0.625 = 0.62.
Usage of Dimensional Analysis
Rounding Off
(i) To check the correctness of a given formula.
Rules for rounding off the numbers:
(ii) To establish relation between quantities dimensionally.
1. Rule: If the digit to be rounded off is more than 5, then the
(iii) To convert the value of a quantity from one system of
preceding digit is increased by one. e.g., 6.87≈ 6.9
units to other system.
2. Rule: If the digit to be rounded off is less than 5, than the
Limitations of Dimensional Analysis preceding digit is unaffected and is left unchanged. e.g., 3.94
(i) It does not predict the numerical value or number ≈ 3.9
associated with a physical quantity in a relation 3. Rule: If the digit to be rounded off is 5 then the preceding
digit is increased by one if it is odd and is left unchanged if it
e.g., v= u + 1 at & v = u + at
is even. e.g., 14.35 ≈ 14.4 and 14.45 ≈ 14.4
3 5
Both are dimensionally valid. Representation of Errors
(ii) It does not derive any relations involving trigonometric, 1. Mean absolute error is defined as
logarithmic and exponential functions ∆a1 + ∆a2 + ... + ∆an n
∆a
e.g., P = P0e–bt cannot be derived dimensionally.=
2 ∆a = ∑ i
n i =1 n
(iii) It does not give any information about dimensionally
Final result of measurement may be written as:
constants or nature of a quantity (vector/scalar) associated
with a relation. a = am ± ∆a
1
