UNITS AND MEASUREMENT-Notes

Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.

A fundamental unit is defined for a quantity that cannot be expressed in terms of other physical quantities. Such quantities are called fundamental (or base) quantities. Their units are fixed by international agreement so that measurements remain uniform everywhere—from school laboratories to space missions.
In the International System of Units (SI), seven base quantities are recognized.

1. Length — metre (m):
   Length tells us how far apart two points are in space. The metre is the basic unit used to measure distance, size, and separation. Whether we describe the width of a wire or the distance between planets, length measurements ultimately trace back to the metre.
   
   It is defined by taking the fixed numerical value of the speed of light in vacuum \(c\) to be 299792458 when expressed in the unit \(m s^{–1}\), where the second is defined in terms of the caesium frequency \(\Delta \nu cs.\)
2. Mass — kilogram (kg):
   Mass measures the amount of matter in an object and reflects its resistance to change in motion. Unlike weight, mass does not depend on gravity. In physics, calculations involving momentum, energy, and force all rely on the kilogram.
   
   It is defined by taking the fixed numerical value of the Planck constant h to be \(6.62607015×10^{–34}\) when expressed in the unit \(J s\), which is equal to \(kg m^2 s^{–1}\), where the metre and the second are defined in terms of \(c\) and \(\Delta \nu cs\).
3. Time — second (s):
   Time measures the sequence and duration of events. The second provides a precise way to quantify intervals—how long a process lasts or how frequently something repeats. Motion, oscillations, and waves all depend critically on time measurement.
   
   It is defined by taking the fixed numerical value of the caesium frequency \(\Delta cs\), the unperturbed ground state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit \(Hz\), which is equal to \(s^{–1}\)
4. Electric Current — ampere (A):
   Electric current measures the rate at which electric charge flows. The ampere forms the backbone of electrical and electronic measurements, linking electricity with magnetism and energy transfer.
   
   It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634×10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of \(\Delta \nu cs\)
5. Thermodynamic Temperature — kelvin (K):
   Temperature indicates the degree of hotness or coldness of a system and is directly related to the average kinetic energy of particles. The kelvin scale begins at absolute zero, making it especially suitable for scientific work.
   
   It is defined by taking the fixed numerical value of the Boltzmann constant \(k\) to be \(1.380649×10^{–23}\) when expressed in the unit \(J K^{–1}\), which is equal to \(kg m^2 s^{–2} k^{–1}\), where the kilogram, metre and second are defined in terms of \(h,\ c\) and \(\Delta \nu cs\)
6. Amount of Substance — mole (mol):
   The mole counts the number of elementary entities (atoms, molecules, ions, etc.) in a substance. It connects the microscopic world of particles to the macroscopic quantities we can measure in the laboratory.
   
   One mole contains exactly \(6.02214076×10^{23}\) elementary entities. This number is the fixed numerical value of the Avogadro constant, \(N_A\), when expressed in the unit \(mol^{–1}\) and is called the Avogadro number. The amount of substance, symbol \(n\), of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.
7. Luminous Intensity — candela (cd):
   Luminous intensity measures the brightness of a light source in a given direction, taking into account the sensitivity of the human eye. It is essential in areas involving illumination and optics.
   
   It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency \(540×10^{12} Hz,\ K_{cd}\), to be 683 when expressed in the unit \(lm W^{–1}\), which is equal to \(cd\ sr\ W^{–1}\), or \(cd\ sr\ kg^{–1}m^{–2}s^3\), where the kilogram, metre and second are defined in terms of \(h,\ c\) and \(\Delta \nu cs\).

A derived unit is a unit obtained by combining fundamental units through multiplication or division, according to the physical relation of the quantity involved.
The quantities measured using such units are called derived quantities.

In simple words, if a quantity can be expressed using base quantities like length, mass, and time, then its unit is a derived unit.

How derived units are formed

Derived units are constructed using algebraic combinations of SI base units.
For example:

• If a quantity involves length divided by time, its unit will involve metre per second.
• If a quantity involves mass multiplied by acceleration, its unit will combine kilogram, metre, and second.

This method keeps all measurements internally consistent within the SI system.

Common Derived Quantities and their Units

CGS System (Centimetre–Gram–Second)

The CGS system is a system of units in which centimetre (cm) is the unit of length, gram (g) is the unit of mass, and second (s) is the unit of time. It was widely used in early scientific studies, especially in mechanics and electromagnetism, but is now largely replaced by the SI system in modern physics.

FPS System (Foot–Pound–Second)

The FPS system is a system of units where foot (ft) is used for length, pound (lb) for mass, and second (s) for time. This system was mainly used in engineering practices in some countries and everyday measurements but is not suitable for scientific work due to its non-decimal nature.

MKS System (Metre–Kilogram–Second)

The MKS system is a system of units in which metre (m) is the unit of length, kilogram (kg) is the unit of mass, and second (s) is the unit of time. It provided a more practical and decimal-based framework and later became the foundation of the modern International System of Units (SI).

International System of Units

The International System of Units (SI) is a standardised system of measurement adopted internationally for scientific, industrial, and everyday use. It provides a common framework so that measurements made in different countries, laboratories, or time periods remain consistent, comparable, and reliable.

Base units of the SI system

The SI system defines seven fundamental quantities, each with a unique base unit. These base units form the foundation of all physical measurements.

SI prefixes

Physical quantities often involve very large or very small values. To handle this conveniently, the SI system uses prefixes to represent powers of ten.

• Kilo (\(\mathrm{k})=10^3\)
• centi (\(\mathrm{c})=10^{-2}\)
• milli (\(\mathrm{m})=10^{-3}\)
• micro (\(\mathrm{\mu})=10^{-6}\)
• mega (\(\mathrm{M})=10^6\)

Rules for writing SI units

To maintain uniformity, the SI system follows strict conventions:

• Unit symbols are written in lowercase letters, except when named after a person (e.g., N, J).
• Symbols are not pluralised and do not end with a full stop.
• A space is left between the numerical value and the unit symbol (e.g., 5 m, 10 kg).

These rules prevent ambiguity and ensure clarity in scientific writing.

When we measure any physical quantity—like length, mass, or time—the result is never perfectly exact. Every measuring instrument has a limited precision, and because of this, the measured value carries some uncertainty. Significant figures are used to clearly show how precise a measurement is.

What are Significant Figures?

Significant figures are the meaningful digits in a measured quantity.
They include:

• All the certain digits, which are known with confidence
• The first uncertain (estimated) digit

In simple words, significant figures tell us which digits in a number are reliable and which digit is the last one that is only an estimate.

Example to Understand the Idea

Suppose you measure the length of a pencil using a scale and find it to be 12.4 cm.

• The digits 1 and 2 are certain
• The digit 4 is estimated (because the scale has a smallest division of 1 mm)

So, 12.4 cm has three significant figures.

Rules for Counting Significant Figures

1. All non-zero digits are significant
   Example:
   
   • 234 \(\Rightarrow\) 3 significant figures
   • 7.81 \(\Rightarrow\) 3 significant figures
2. Zeros between non-zero digits are significant
   Examples:
   
   • 101 \(\Rightarrow\) 3 significant figures
   • 2.05 \(\Rightarrow\) 3 significant figures
3. Leading zeros are not significant
   (zeros before the first non-zero digit only locate the decimal point)
   Examples:
   
   • 0.0025 \(\Rightarrow\) 2 significant figures
   • 0.040 \(\Rightarrow\) 2 significant figures
4. Trailing zeros after the decimal point are significant
   Examples:
   
   • 2.300 \(\Rightarrow\) 4 significant figures
   • 0.0600 \(\Rightarrow\) 3 significant figures
5. Trailing zeros in whole numbers without a decimal point are not clearly significant
   Examples:
   
   1500 \(\Rightarrow\) may have 2, 3, or 4 significant figures (ambiguous)

   To avoid confusion, scientific notation is used.

Scientific Notation and Significant Figures

Scientific notation clearly shows the number of significant figures.
Examples:

• \(1.50\times 10^3 \Rightarrow 3\) significant figures
• \(1500\times 10^3 \Rightarrow 4\) significant figures

Important of Significant Figure

Significant figures are important because they:

• Indicate the precision of measurement
• Prevent writing misleadingly accurate results
• Ensure correct rounding in calculations
• Help maintain consistency in experimental results

In physics, writing extra digits that are not justified by measurement accuracy is considered incorrect.

Significant Figures in Calculations (NCERT Rule)

• Addition or subtraction:
  The result should have the same number of decimal places as the quantity with the least decimal places.
• Multiplication or division:
  The result should have the same number of significant figures as the quantity with the least significant figures.

Rounding off is the process of reducing the number of digits in a numerical value while keeping it as close as possible to the original value.
It is done so that the final result matches the precision of the measured data and does not give a false sense of accuracy.

In physics, rounding off is necessary because no measurement is exact.

Rules of Rounding Off

Suppose a number is to be rounded to a certain number of significant figures.

• Rule 1:
  If the digit to be dropped is less than 5
  The preceding digit remains unchanged.
  Example
  \(2.743\approx 2.74\) (rounded to 3 significant figures)
• Rule 2:
  If the digit to be dropped is greater than 5
  The preceding digit is increased by 1.
  Example
  \(5.786\approx 5.79\)
• Rule 3:
  If the digit to be dropped is exactly 5
  
  • If the preceding digit is even, it is left unchanged
  • If the preceding digit is odd, it is increased by 1

  This rule helps avoid systematic rounding errors.

  Example:
  \[2.3\color{orange}{4}\color{white}5\approx 2.34 \text{ (4 is even)}\] \[2.3\color{orange}{5}\color{white}5\approx 2.36 \text{ (5 is odd)}\]

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

Side of cube =7.203 m
Signnificant figure in measurement of side of cube is 4 digit, therfore, significant digit in volume of the surface area and volume will be rounded off to 4 signifivan figure
SA of cube = \[\begin{aligned}SA&=6(\text{side})^2\\ &=6\times 7.203^2\\ &=6\times 7.203\times 7.203\\ &=311.299254\\ &=311.3\ m^2\end{aligned}\] Volume of cube = \[\begin{aligned}V&=(\text{side})^3\\ &=7.203^3\\ &=7.203\times 7.203\times 7.203\\ &=373.714754427\\ &=373.7\ m^3\end{aligned}\]

5.74 g of a substance occupies 1.2 \(\mathrm{cm^3}\). 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 \(d\)=

\[\begin{aligned}d&=\dfrac{\text{Mass}}{\text{Volume}}\\\\ &=\dfrac{5.74}{1.2}\\\\ &=4.7833333333\\ &=4.8\ \mathrm{g\ cm^{-3}} \end{aligned}\]

The dimensions of a physical quantity are the powers to which the fundamental physical quantities must be raised in order to represent that quantity.

In mechanics, the three fundamental quantities are:
Length → [L]
Mass → [M]
Time → [T]
Every physical quantity can be expressed as a combination of these basic dimensions.

The dimensional formula of a physical quantity is an expression that shows how the quantity depends on the fundamental quantities.
If a physical quantity \(Q\) depends on length \(L\), mass \(M\), and time \(T\) as:

\[Q=M^aL^bT^c\]

then its dimensional formula is:

\[[Q]=[M^aL^bT^c]\]

Here, \(a,\ b, \text{ and } c\) are called the dimensional exponents.

Examples of Dimensional Formulae

• Velocity:
  \(\text{Velocity}=\dfrac{\text{Distance}}{\text{Time}}\) \[[v]=[LT^{-1}]\]
• Acceleration:
  \(\text{Acceleration}=\dfrac{\text{change in velocity}}{\text{time}}\) \[[a]=[LT^{-2}]\]
• Force:
  \(\text{Force} = \text{mass} \times \text{acceleration}\) \[[F]=[MLT^{-2}]\]

Dimensional Constants and Dimensionless Quantities

• A dimensional constant has dimensions, such as gravitational constant or Planck’s constant.
• A dimensionless quantity has no dimensions, even though it may have a unit.

Examples of dimensionless quantities:

• Strain
• Angle (radian)
• Coefficient of friction
• Refractive index

The principle of homogeneity of dimensions states that in any correct physical equation, the dimensions of all terms on the left-hand side must match exactly those on the right-hand side. This ensures the equation makes physical sense regardless of the unit system used. It forms a core part of dimensional analysis

• Checking the correctness of equations:
  An equation is dimensionally correct only if the dimensions of the left-hand side and right-hand side are the same.
• Deriving relations between physical quantities:
  Dimensional analysis helps in finding how one quantity depends on others when the exact relation is unknown.
• Converting units from one system to another:
  By comparing dimensions, we can convert a quantity from CGS to SI or any other system of units.

Let us consider an equation \(\frac{1}{2}m v^2= m g h\) 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

Equation: \(\frac{1}{2}m v^2= m g h\)

Considering LHS of equation:
\[\dfrac{1}{2}mv^2\] Dimension of Mass \(m\)=\([M]\)
Dimension of Squared Velocity \(v^2\) = \([L^2T^{-2}]\) Combining both dimensions LHS becomes

\[[M][L^2T^{-2}]=[ML^2T^{-2}]\tag{1}\]

Consider RHS Dimension of Mass \(m\) =\([M]\)
Dimension of Acceleration due to gravity \(g\)= \([LT^{-2}]\)
Dimension of height \(h\)=\([L] \)
Combinig all three, we get

\[[M][LT^{-2}][L]=[ML^2T^{-2}]\tag{2}\] LHS=RHS, Hence equation is dimensionally correct.

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

\(T\) is dependent on \(l\, m\ \text{ g} ), hence we can write dependency as

\[ \begin{align} T&\propto l^a\ m^b\ g^c]\\ T&=k\ l^a\ m^b\ g^c\tag{1}\\ \end{align} \]

Dimension of \(l\)=\([L]\)
Dimension of \(m\)=\([M]\)
Dimension of \(g\)=\([LT^{-2}]\)
Dimension of \(T\)=\([T]\)

Substituting dimensional values in equation (1)

\[ \begin{align} T&=k\ l^a\ m^b\ g^c\\\\ [L^0\ M^0\ T^1]&=[L^a]\ [M^b]\ [(L\ T^{-2})^{c}]\\\\ [L^0\ M^0\ T^1]&=[L^a]\ [M^b]\ [{L^c\ T^{-2c}}]\\\\ [L^0\ M^0\ T^1]&=[L^{a+c}\ M^b\ {T^{-2c}}]\\\\ \end{align} \]

comparing power of \(M,\ L \text{ and } T\) from both side of equation

\[ \begin{aligned} b&=0\\ a+c&=0\\ -2c&=1\\ \Rightarrow c&=-\dfrac{1}{2}\\\\ a+(-\dfrac{1}{2})&=0\\\\ \Rightarrow a&=\dfrac{1}{2} \end{aligned} \]

Substituting values of \(a,\ b,\ \text{and}\ c\) in equation-(1)

\[ \begin{aligned} T&=k\ l^a\ m^b\ g^c\\ &=kl^{\frac{1}{2}}\ m^0\ g^{-\frac{1}{2}}\\\\ &=k\sqrt{\dfrac{l}{g}} \end{aligned} \]

\(k=2\pi\), hence,

\[T=2\pi\sqrt{\dfrac{l}{g}}\]

• Physics is a quantitative science, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities (such as length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity).
• Each base quantity is defined in terms of a certain basic, arbitrarily chosen but properly standardised reference standard called unit (such as metre, kilogram, second, ampere, kelvin, mole and candela). The units for the fundamental or base quantities are called fundamental or base units.
• Other physical quantities, derived from the base quantities, can be expressed as a combination of the base units and are called derived units. A complete set of units, both fundamental and derived, is called a system of units.
• The International System of Units (SI) based on seven base units is at present internationally accepted unit system and is widely used throughout the world.
• The SI units are used in all physical measurements, for both the base quantities and the derived quantities obtained from them. Certain derived units are expressed by means of SI units with special names (such as joule, newton, watt, etc).
• The SI units have well defined and internationally accepted unit symbols (such as m for metre, kg for kilogram, s for second, A for ampere, N for newton etc.).
• Physical measurements are usually expressed for small and large quantities in scientific notation, with powers of 10. Scientific notation and the prefixes are used to simplify measurement notation and numerical computation, giving indication to the precision of the numbers.
• Certain general rules and guidelines must be followed for using notations for physical quantities and standard symbols for SI units, some other units and SI prefixes for expressing properly the physical quantities and measurements.
• In computing any physical quantity, the units for derived quantities involved in the relationship(s) are treated as though they were algebraic quantities till the desired units are obtained.
• In measured and computed quantities proper significant figures only should be retained. Rules for determining the number of significant figures, carrying out arithmetic operations with them, and ‘rounding off ‘ the uncertain digits must be followed.
• The dimensions of base quantities and combination of these dimensions describe the nature of physical quantities. Dimensional analysis can be used to check the dimensional consistency of equations, deducing relations among the physical quantities, etc. A dimensionally consistent equation need not be actually an exact (correct) equation, but a dimensionally wrong or inconsistent equation must be wrong.