Units and Measurements

Fundamental Concepts of Units and Measurements

• Definition of Unit: The measurement of any physical quantity involves a comparison with a certain basic, internationally accepted reference standard called a Unit.
• Expression of Physical Quantity: Any physical quantity is expressed as a combination of a numerical value and a unit.
• Classification of Physical Quantities:
  • Fundamental Quantity: Basic quantities that are independent of others.
  • Derived Quantity: Quantities expressed in terms of fundamental quantities.

Systems of Units

• Three Systems of Units:
  1. CGS System: Centimeter, Gram, Second.
  2. FPS System: Foot, Pound, Second.
  3. MKS System (SI): Meter, Kilogram, Second.
• Dimensionless Quantities and Their Units:
  • Plane Angle: The unit is the radian ($rad$).
  • Solid Angle: The unit is the steridian ($sr$).

SI Base Quantities

• The International System of Units (SI) defines seven base quantities:
  • Length: metre ($m$)
  • Mass: kilogram ($kg$)
  • Time: second ($s$)
  • Electric Current: ampere ($A$)
  • Temperature: Kelvin ($K$)
  • Amount of Substance: mole ($mol$)
  • Luminous Intensity: candela ($cd$)

Measurement of Length

• Measurement Tools and Accuracy:
  • Vernier Callipers: Used for measuring lengths to an accuracy of $10^{-4}\,m$.
  • Screw Gauge: Used for measuring lengths to an accuracy of $10^{-5}\,m$.
• Measurement of Large Distances (Parallax Method):
  • Basis: The distance between two points of observation is called the basis (denoted as $b$).
  • Parallax Angle: Let $S$ be a distant object and $A$ and $B$ be two observation points. The angle $\angle ASB$ is the parallax angle (denoted as $\theta$).
  • Calculation: As the distance $D$ is very large, $\frac{b}{D} << 1$. Therefore, $\theta$ is very small. We can approximately take $AB$ as an arc of length $b$ of a circle with center $S$ and radius $D$.
  • Formula: $D = \frac{b}{\theta}$

Estimation of Very Small Distances (Size of Molecules)

• Need for Special Equipment: Measuring molecular sizes requires special instruments like electron microscopes.
• Electron Microscope Resolution: The resolution is limited by the fact that electrons behave as waves.
• Wavelength and Resolution:
  • The wavelength of an electron is less than $1\,\text{\AA}$.
  • Electron microscopes with a resolution of $0.6\,\text{\AA}$ have been built.
  • These instruments can resolve individual atoms and molecules within a material.

Measurement of Mass

• Standard Prototypes: Prototypes of the international standard kilogram are supplied by the International Bureau of Weights and Measures (IBWM).
• Unified Atomic Mass Unit ($u$): This is the standard unit for mass at the atomic and molecular scale.
  • Definition: $1\,u = \frac{1}{12} \times \text{mass of an atom of Carbon-12}$.
  • Value: $1\,u = 1.66 \times 10^{-27}\,kg$.

Measurement of Time

• Cesium Atomic Clock: Time is measured based on the transition of the Cesium atom between hyperfine levels of its ground state.
  • Definition of 1 Second: Exactly $9,192,631,770$ vibrations of the Cesium-133 atom.
• Accuracy: The accuracy of these clocks is approximately $1 \times 10^{-13}$ per year.
• Uncertainty: The gain or loss of time in a Cesium clock is not more than $3\,\mu s$ per year.

Accuracy, Precision, and Errors in Measurement

• Error: The uncertainty inherent in any measurement is called an error.
• Accuracy: A measure of how close the measured value is to the true value of the quantity.
• Precision: Describes the closeness of two or more measurements to each other. Precision depends primarily on the resolution of the measuring instrument.
• Example Comparison:
  • True value of an object's length: $6.39\,m$.
  • Case 1: Device with resolution $0.1\,m$ yields $6.28\,m$. This is more accurate (closer to true value) but not precise due to lower resolution.
  • Case 2: Device with resolution $0.01\,m$ yields $6.17\,m$. This is more precise but not accurate.

Types of Errors

Systematic Errors

• Instrumental Errors: Arise from faults in the instrument, such as imperfect design or zero errors. (e.g., in Vernier callipers, the zero mark of the scale does not coincide with the zero mark of the main scale).
• Imperfection in Experimental Technique: For example, placing a thermometer under the armpit to check body temperature will show a value lower than the actual temperature due to procedural limits.
• Personal Errors: Arise from individual bias, such as careless viewpoint, incorrect instrument usage, or general carelessness.
• Minimization: Systematic errors can be minimized by improving experimental techniques.

Random Errors

• Arise due to random and unpredictable fluctuations in experimental conditions (e.g., temperature changes).

Least Count Error

• Definition: The smallest value that can be measured by a measuring instrument.
• Nature: It is an error associated with the resolution of the instrument and belongs to the category of random errors within a limited size.
• Minimization: Can be reduced by using instruments with higher precision, improving experimental techniques, repeating observations several times, and calculating the arithmetic mean.

Mathematical Analysis of Errors

• Absolute Error: The magnitude of the difference between the individual measurement and the true value.
  • Notation: $|\Delta a|$
  • Calculation: If the true value is not given, the arithmetic mean ($a_{mean}$) is used as the true value. The error is the difference between the individual measurement and the mean. Absolute error is always taken as positive.
• Mean Absolute Error (\Delta a_{mean}): The mean of all absolute errors is taken as the final absolute error of the physical quantity.
• Range of Value: The final result is expressed as $a = a_{mean} \pm \Delta a_{mean}$, meaning the value lies between $(a_{mean} - \Delta a_{mean})$ and $(a_{mean} + \Delta a_{mean})$.
• Relative Error: The ratio of the mean absolute error to the mean value of the quantity measured.
  • Formula: $\text{Relative error} = \frac{\Delta a_{mean}}{a_{mean}}$
• Percentage Error: Expression of relative error in percentage.
  • Formula: $\text{Percentage error} = \frac{\Delta a_{mean}}{a_{mean}} \times 100\%$

Combination of Errors

• Errors in Sum and Difference:
  • Let $Z = A + B$ or $Z = A - B$.
  • Measured values: $A \pm \Delta A$ and $B \pm \Delta B$.
  • Rule: When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors of the individual quantities.
  • Max Error: $\Delta Z = \Delta A + \Delta B$
• Errors in Product and Quotient:
  • Let $Z = ab$.
  • $Z \pm \Delta Z = (a \pm \Delta a)(b \pm \Delta b) = ab \pm a\Delta b \pm b\Delta a \pm \Delta a\Delta b$
  • Dividing by $Z = ab$ on both sides: $1 \pm \frac{\Delta Z}{Z} = 1 \pm \frac{\Delta b}{b} \pm \frac{\Delta a}{a} + \frac{\Delta a\Delta b}{ab}$.
  • Since $\Delta a$ and $\Delta b$ are small, the term $\frac{\Delta a\Delta b}{ab}$ is negligible.
  • Rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors of the individual quantities.
  • Max Relative Error: $\frac{\Delta Z}{Z} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$.

Significant Figures

• Definition: The reliable digits plus the first uncertain digit in a measurement are known as significant figures.
• General Rules:
  • A change in units does not change the number of significant digits.
  • All non-zero numbers are significant.
  • All zeros between two non-zero numbers are significant.
  • If a number is less than 1, zeros to the right of the decimal point but left of the first non-zero digit are not significant (e.g., in $0.0028$, there are 2 significant figures).
  • Terminal zeros in a number without a decimal point are not significant.
  • Terminal zeros in a measurement with a decimal point are significant.

Scientific Notation and Arithmetic Operations

• Scientific Notation: Numbers are expressed as $b \times 10^a$, where $10^a$ is called the order of magnitude.
• Significance in Notation: The power of 10 is irrelevant for determining significant figures. All zeros appearing in the base number ($b$) are significant.
• Arithmetic Rules:
  • Multiplication and Division: The final result should retain as many significant figures as the original number with the least significant figures.
  • Addition and Subtraction: The final result should retain as many decimal places as there are in the original number with the least decimal places.

Dimensions of Physical Quantities

• Definition: Dimensions are the powers to which base quantities are raised to specify a physical quantity. Square brackets $[ ]$ are used to represent dealing with dimensions.
• Checking Dimensional Consistency: Physical quantities can be added or subtracted only if they have the same dimensions. This is known as the Principle of Homogeneity.