“Units and Measurements: Dimensions, Errors and Significant figures explained”

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

  1. Introduction:

     Physics is a quantitative science that relies on the measurement of physical quantities. Every experiment in physics involves observing, measuring, and comparing quantities such as length, mass, time, and temperature with internationally accepted standards.

  1.1. Unit of a quantity:

     A measurement is always a comparison between an unknown quantity and a standard quantity of the same kind.

“The standard measure of any physical quantity is called the unit of that quantity.”

     For example, to measure the mass of a fruit, we compare it with standard mass units such as 1 kg or 500 g.

 2. Units and Measurements:

  2.1. Requirements of a good unit:

A good unit of measurement must satisfy the following conditions:

1. Invariable: Its value should not change with time, place, or physical conditions.
2. Reproducible: It should be possible to reproduce the unit accurately anywhere in the world.
3. Universally accepted: The unit should be accepted internationally.
4. Accessible: It should be easy to use for comparison.

  2.2. Classification of Physical quantities:

     Physical quantities studied in physics are broadly classified into two types:

  2.2.1. Fundamental physical quantities:

   “A fundamental quantity is a physical quantity that does not depend on any other quantity.”

    Examples include length, mass, time, and temperature.

   There are seven fundamental physical quantities, and the units used to measure them are called fundamental units.

  2.2.2. Derived physical quantities:

     “A derived quantity is a physical quantity that depends on one or more fundamental quantities.”

    Examples include velocity, acceleration, force, work, pressure, and density.

    The units used to measure derived quantities are known as derived units.

2.3. Systems of Units:

      “A collection of units for measuring physical quantities is called a system of units”.

   Common systems of units are:

    FPS system:

       Foot, Pound, Second

     CGS system:

        centimetre, gram, second

    MKS system:

         metre, kilogram, second

    SI system:

    International System of Units (Système International d’Unités)

   2.4. Seven SI Fundamental units:

Physical quantity	SI Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

 2.5. Supplementary Units:

Although no longer classified separately in SI, the following are commonly used:

Quantity	Unit	Symbol
Plane angle	radian	rad
Solid angle	steradian	sr

 2.6. Conventions for the Use of SI Units:

1. The full name of a unit always begins with a lowercase letter, even if it is named after a scientist (e.g., newton, joule).
2. The symbol of a unit named after a person is written with a capital letter (e.g., N for newton, J for joule).
3. Symbols of other units are written in lowercase letters (e.g., m for metre, s for second).
4. Unit symbols are never written in plural form (e.g., 25 m, not 25 ms).
5. No full stop or punctuation mark is used after a unit symbol.
6. Every physical quantity should be represented using its proper unit symbol.

    ( e.g. mass of ball= 2 kg )

  3. Dimensions and Dimensional formula:

    3.1. Definition of Dimensions:

  “The dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain the unit of that quantity.”

   When any derived quantity is expressed with appropriate powers of symbols of the fundamental quantities, then such an expression is called dimensional formula.       

      Dimensions are expressed in square brackets using symbols for fundamental quantities:

 Mass → M

 Length → L

 Time → T

    A general dimensional formula is written as:

[ Mᵃ Lᵇ Tᶜ ]

   3.2. Example: Dimensions of Velocity:

1. Formula:

   Velocity = Displacement / Time

2. Symbolic form:

         v = L / T

3. Dimensional formula:

       [ v ] = [ M⁰ L¹ T⁻¹ ]

  3.3. Dimensional formulae of some physical quantities:

Sr. No.	Physical Quantity	Formula	SI Unit	Dimensional Formula
1	Velocity	Velocity=displacement/time	m/s	[ M⁰ L¹ T⁻¹ ]
2	Volume	Volume=cube of length	m3	[ M⁰ L3 T0 ]
3	Density	Density=mass/volume	kg/m3	[ M1 L-3 T0 ]
4	Acceleration	Acceleration=velocity/time	m/s2	[ M⁰ L¹ T⁻2 ]
5	Momentum	Momentum=Mass x Velocity	kg m/s	[ M1 L¹ T⁻¹ ]
6	Force	Force= mass x acceleration	N	[ M1 L¹ T⁻2 ]
7	Work or Energy	Work =Force x Displacement	J	[ M1 L2 T⁻2 ]
8	Power	Power=Work/time	W	[ M1 L2 T⁻3 ]
9	Torque	τ =rFsinϴ	Nm or J	[ M1 L2 T–2 ]
10	Moment of Inertia	M.I = mr2	kg m2	[ M1 L2 T0 ]
11	Frequency	n =cycles/second	Hz	[ M⁰ L0 T⁻¹ ]
12	Wavelength	λ = Length	m	[ M⁰ L¹ T0 ]

  3.4. Applications of Dimensional analysis:

    Dimensional analysis is useful for:

1. Checking the correctness of physical equations.

     “An equation involving different physical quantities is dimensionally correct if the dimensions of every term on both sides of the equation are identical.”

   Consider first equation of motion :

     v = u + at

   Dimensions of [v] =[M⁰ L¹ T^-1]  =[u]

    [at] =[M⁰ L¹ T^-2]  [M⁰ L⁰ T¹]  =[ M⁰ L¹ T^-1] 

   Hence, dimensions of   [L.H.S.] = [R.H.S.]

   Thus, this equation is dimensionally correct.

2. Converting units from one system to another.
3. Deriving relationships between physical quantities.

   3.5. Limitations of Dimensional analysis:

1. It cannot determine numerical (dimensionless) constants.
2. It cannot be applied to equations involving trigonometric, exponential, or logarithmic functions.
3. It fails when the proportionality constant has dimensions.
4. It cannot identify additional terms having the same dimensions.

 4. Errors in measurements:

      In experimental physics, the measured value of a quantity may differ from its true value due to uncertainties known as errors.

 4.1. Causes of incorrect results:

1. Mistakes – Caused by human negligence; these can be avoided.
2. Errors – Uncertainties inherent in measurement; these cannot be eliminated but can be minimized.

 4.2. Types of Errors:

1. Instrumental Errors:

   Errors caused due to faulty construction or calibration of instruments (e.g., improperly graduated thermometer).

2. Systematic Errors:

    Errors arise due to defective experimental setup (e.g., an ammeter not reading zero when no current flows).

3. Personal Errors:

   Errors introduced by the observer, such as parallax error while reading a scale.

4. Random Errors:

    Errors caused by unpredictable variations in experimental conditions like temperature, pressure, or voltage.

  4.3. Minimization of Errors:

 Errors can be minimized by:

1. Measuring large magnitudes of quantities.
2. Taking multiple readings and calculating their mean.
3. Using instruments with the smallest possible least count.

 4.4. Estimation of Errors:

  Absolute Error:

     “The difference between the measured value and the mean value of a quantity is called absolute error.”

   If a₁, a₂, a₃, …, a_n are measured values, then the mean value is:

         a_m = (a₁ + a₂ + a₃ + … + a_n) / n

   Absolute error:

Δaᵢ = a_m − aᵢ

   Mean absolute error:

Δa_mean = ( |Δa₁| + |Δa₂| + … + |Δa_n| ) / n

    Relative error:

Relative error = Δ_amean / a_m

    Percentage error:

Percentage error = (Δa_mean / a_m) × 100%

  4.5. Propagation of errors:

     Propagation of errors (or propagation of uncertainty) is the mathematical process used to determine how the uncertainties in individual measurements affect the uncertainty of a final result calculated from those measurements.

        In simpler terms, if you measure two different things (like length and width) and each has a small mistake or uncertainty, the final value you calculate (like area) will also have an uncertainty that depends on the errors of the original two measurements.

    Addition or Subtraction:

If A = a ± b, then:

ΔA = Δa + Δb

   Multiplication or Division:

If P = ab or P = a / b, then:

ΔP / P = (Δa / a) + (Δb / b)

   Power rule:

If Q=a_n, then

ΔQ/Q=nΔa/a

  5. Significant figures:

     “The number of digits in a measured value that are known with certainty and plus one uncertain digit are called significant figures.”

 5.1.  Rules for Significant figures:

1. All non-zero digits are significant.

                e.g. Let mass= 345.57 kg, then the significant figure is 5.

2. Zeros between non-zero digits are significant.

                e.g.  Consider, length = 2.05 m, then the significant figure is 3.

3. Zeros to the left of the first non-zero digit are not significant.

              e.g.  If, time = 0.00 624 s, then the significant figure is 3.

4. Zeros to the right of a non-zero digit in the decimal part are significant.

         e.g. Let, volume = 38.23000 m³ , then the significant figure is 7.

 5.2. Significant figures in calculations:

     Addition or Subtraction:

    The result should retain the least number of decimal places among the quantities.

     Multiplication or Division:

    The result should contain the least number of significant figures among the quantities.

 5.3. Rounding off Significant figures:

1. If the digit to be dropped is less than 5, the preceding digit remains unchanged.
2. If the digit to be dropped is greater than 5, the preceding digit is increased by one.
3. If the digit to be dropped is 5 followed by non-zero digits, the preceding digit is increased by one.
4. If the digit to be dropped is exactly 5 or 5 followed by zeros, the preceding digit is rounded to the nearest even number.

 6. Conclusion:

    The concepts of units and measurements, dimensions, errors, and significant figures form the foundation of experimental physics. A clear understanding of these topics is essential for accurate measurements, reliable calculations, and meaningful interpretation of experimental results.

    Mastery of this unit not only strengthens conceptual clarity but also enhances problem-solving skills required for higher studies and competitive examinations.