Units and Measurement Study Notes

Introduction to Units and Measurement

Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called a unit. The result of such a measurement is expressed by a number (numerical measure) accompanied by a unit.

Fundamental and Derived Units

1. Fundamental or Base Quantities: A limited number of units are required to express all physical quantities because they are inter-related. These units are called fundamental or base units.

2. Derived Quantities: Units that are expressed as combinations of base units are known as derived units.

3. System of Units: A complete set, consisting of both base and derived units, is known as a system of units.

The International System of Units (SI)

Historically, different countries used different systems:

• CGS System: Centimetre, gram, and second.

• FPS (British) System: Foot, pound, and second.

• MKS System: Metre, kilogram, and second.

Currently, the internationally accepted standard is the Système Internationale d’ Unites (French for International System of Units), abbreviated as SI. It was developed and recommended by the General Conference on Weights and Measures in 1971 for scientific, technical, industrial, and commercial usage. SI is based on a decimal system, making conversions convenient.

SI Base Units and Definitions

Supplementary Dimensionless Units

1. Plane Angle ($d\theta$): The ratio of the length of arc $ds$ to the radius $r$. The unit is radian ($rad$).

2. Solid Angle ($d\Omega$): The ratio of the intercepted area $dA$ of a spherical surface to the square of its radius $r^2$. The unit is steradian ($sr$).

Measurement of Length

Direct and Indirect Methods

• Metre Scale: $10^{-3}\,m$ to $10^{2}\,m$.

• Vernier Callipers: Accuracy of $10^{-4}\,m$.

• Screw Gauge / Spherometer: Measure down to $10^{-5}\,m$.

Parallax Method for Large Distances

To measure distances of stars or planets, the parallax method is used. Parallax is the apparent shift in the position of an object against a background when viewed from two different positions. The distance between the observation points is the basis ($b$).

For a distant planet $S$ observed from points $A$ and $B$ on Earth: $D = \frac{b}{\theta}$ Where:

• $D$ = Distance to the planet.

• $b$ = Distance between observation points ($AB$).

• $\theta$ = Parallactic angle (in radians).

Angular Size ($\alpha$): If $d$ is the diameter of a planet and $\alpha$ is its angular size subtended at Earth: $\alpha = \frac{d}{D}$

Estimation of Very Small Distances: Molecular Size

Optical microscopes are limited by the wavelength of visible light ($4000\,Å$ to $7000\,Å$; $1\,Å = 10^{-10}\,m$). Electron microscopes use electron beams behaving as waves and can reach resolutions of $0.6\,Å$.

Oleic Acid Experiment: Oleic acid is a soapy liquid with molecular size approx $10^{-9}\,m$.

1. Dissolve $1\,cm^3$ of oleic acid in alcohol to make $20\,cm^3$ solution.

2. Dilute $1\,cm^3$ of this solution again to $20\,cm^3$. Concentration = $\frac{1}{20 \times 20}\,cm^3$ of oleic acid per $cm^3$ of solution.

3. Drop $n$ drops (each of volume $V$) onto water surface sprinkled with lycopodium powder. The film forms a circular area $A$.

4. Thickness ($t$): $t = \frac{\text{Volume of film}}{\text{Area of film}} = \frac{nV}{A \times 20 \times 20}$ If the film is one molecule thick, $t$ represents the molecular diameter.

Range of Lengths

• Tiny nucleus: $10^{-14}\,m$.

• Observable universe: $10^{26}\,m$.

• Special Units:

  • $1\text{ fermi } (1\,f) = 10^{-15}\,m$

  • $1\text{ angstrom } (1\,Å) = 10^{-10}\,m$

  • $1\text{ astronomical unit } (1\,AU) = 1.496 \times 10^{11}\,m$

  • $1\text{ light year } (1\,ly) = 9.46 \times 10^{15}\,m$

  • $1\text{ parsec } = 3.08 \times 10^{16}\,m$

Measurement of Mass

Mass is an intrinsic property of matter independent of temperature, pressure, or location.

• SI Unit: Kilogram ($kg$). In India, the standard is maintained at the National Physical Laboratory (NPL), New Delhi.

• Unified Atomic Mass Unit ($u$): Used for atoms.     $1\,u = \frac{1}{12} \text{ of the mass of a carbon-12 atom (including electrons)} = 1.66 \times 10^{-27}\,kg$

• Ranges: Electron ($10^{-30}\,kg$) to Universe ($10^{55}\,kg$).

• Methods: Common balance for groceries; Gravitational method for planets; Mass spectrograph for particles (radius of trajectory proportional to mass).

Measurement of Time

Time is measured using clocks based on periodic vibrations. The Cesium Clock (atomic clock) uses vibrations of the cesium-133 atom.

• One second corresponds to $9,192,631,770$ vibrations of the radiation between two hyperfine ground levels of the atom.

• Accuracy: Cesium clocks are so stable that the uncertainty is $1$ part in $10^{13}$, losing or gaining only $3\,\mu s$ per year.

• Coincidence: The ratio of the longest to shortest lengths in the universe is approx $10^{41}$. The ratio for time is also approx $10^{41}$. The ratio for masses is approx $(10^{41})^2$.

Accuracy, Precision, and Errors in Measurement

• Accuracy: How close the measured value is to the true value.

• Precision: The resolution or limit to which the quantity is measured.

• Error: The uncertainty in a measurement.

Classification of Errors

1. Systematic Errors: Tend to be in one direction (positive or negative).

   • Instrumental: Imperfect calibration or zero error.

   • Imperfection in Technique: e.g., measuring body temperature under the armpit gives a lower value than actual.

   • Personal Errors: Observer bias or parallax.

2. Random Errors: Occur irregularly in size and sign due to unpredictable fluctuations (voltage, temperature, vibrations).

3. Least Count Error: The smallest value measurable by an instrument. It is a type of random error but related to resolution.

Mathematical Analysis of Errors

Given measurements $a_1, a_2, ..., a_n$:

• Mean Value: $a_{mean} = \frac{\sum a_i}{n}$

• Absolute Error: $|\Delta a_i| = |a_i - a_{mean}|$

• Mean Absolute Error: $\Delta a_{mean} = \frac{\sum |\Delta a_i|}{n}$

• Relative Error: $\frac{\Delta a_{mean}}{a_{mean}}$

• Percentage Error: $\delta a = \left( \frac{\Delta a_{mean}}{a_{mean}} \right) \times 100\%$

Combination of Errors

• Sum or Difference ($Z = A \pm B$): $\Delta Z = \Delta A + \Delta B$

• Product or Quotient ($Z = AB$ or $Z = A/B$): $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$

• Power ($Z = A^k$): $\frac{\Delta Z}{Z} = k \left( \frac{\Delta A}{A} \right)$

• General Rule ($Z = \frac{A^p B^q}{C^r}$): $\frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}$

Significant Figures

Significant figures comprise all reliably known digits plus the first uncertain digit.

Rules for Determining Significant Figures

1. All non-zero digits are significant.

2. Zeros between non-zero digits are significant.

3. If a number is less than 1, zeros to the right of the decimal but left of the first non-zero digit are NOT significant (e.g., in $0.002308$, underlined zeros are not significant).

4. Trailing zeros in a number without a decimal are NOT significant (e.g., $12300$ has 3 significant figures).

5. Trailing zeros in a number with a decimal ARE significant (e.g., $3.500$ has 4 significant figures).

Rules for Arithmetic Operations

• Multiplication/Division: The final result must have the same number of significant figures as the original number with the least significant figures.

• Addition/Subtraction: The final result must have the same number of decimal places as the number with the least decimal places.

Rounding Off Rules

• If the digit to be dropped is $> 5$, raise the preceding digit by 1.

• If the digit to be dropped is $< 5$, leave the preceding digit unchanged.

• If the digit to be dropped is exactly 5:

  • If preceding digit is even, drop the 5 without change.

  • If preceding digit is odd, raise the preceding digit by 1.

Dimensions of Physical Quantities

Dimensions are the powers to which base quantities ($[M], [L], [T], [A], [K], [cd], [mol]$) are raised to represent a derived quantity.

• Dimensional Formula: Expression showing how base quantities represent a quantity (e.g., Force = $[MLT^{-2}]$).

• Dimensional Equation: Equating a physical quantity to its dimensional formula (e.g., $[F] = [MLT^{-2}]$).

Dimensional Analysis Applications

1. Checking Consistency: Use the Principle of Homogeneity (only quantities with the same dimensions can be added/subtracted).

   • Example: $x = x_0 + v_0 t + \frac{1}{2} a t^2$ is correct because each term is dimensionally length $[L]$.

2. Deducing Relations: Deriving formulas by assuming power-law dependencies.

   • Example: For a pendulum, $T = k l^x g^y m^z$. Solving dimensions gives $T = 2\pi\sqrt{\frac{l}{g}}$.

Examples and Numerical Data

• Example 2.1: $1^{\circ} = 1.745 \times 10^{-2}\,rad$; $1' = 2.91 \times 10^{-4}\,rad$; $1'' = 4.85 \times 10^{-6}\,rad$.

• Example 2.6: Comparing two clocks. Clock 2 is better for precision because its variation over time is lower ($31\,s$ vs $162\,s$), even if its absolute time is less accurate (zero error can be corrected).

• Example 2.12: In determining $g = \frac{4\pi^2L}{T^2}$, the total percentage error is $\frac{\Delta L}{L} + 2\left(\frac{\Delta T}{T}\right)$.

• Example 2.15: Kinetic Energy $K = \frac{1}{2}mv^2$. Dimensional check: $[M][LT^{-1}]^2 = [ML^2T^{-2}]$. Potential Energy $mgh$: $[M][LT^{-2}][L] = [ML^2T^{-2}]$. Equation is dimensionally consistent.