Introduction

Measurement is a fundamental process in physics and science, involving the comparison of a physical quantity with a predefined, internationally agreed-upon reference standard known as a unit.

The result of a measurement is always expressed as a numerical value accompanied by the appropriate unit (e.g., 5 meters, 10 kilograms).

While there are numerous physical quantities, they are interrelated, meaning we only need a limited number of fundamental units to describe all others.

Units associated with the basic or fundamental quantities are called fundamental or base units.

Units for all other physical quantities, which can be expressed as combinations of base units, are called derived units.

A system of units refers to a complete set encompassing both base units and derived units.

The International System Of Units

Historically, different countries used different systems of units. Common examples included:

• CGS System: Based on centimetre (length), gram (mass), and second (time).
• FPS System (British System): Based on foot (length), pound (mass), and second (time).
• MKS System: Based on metre (length), kilogram (mass), and second (time).

The internationally accepted system currently is the Système Internationale d’ Unites, abbreviated as SI.

• It was developed by the Bureau International des Poids et Measures (BIPM).
• It was formally established in 1971 and has undergone revisions, including significant changes in 2018 based on fundamental physical constants.
• SI units are used worldwide in scientific, technical, industrial, and commercial fields.
• A key advantage is its decimal nature, simplifying conversions.

The SI system is built upon seven base units:

The definitions of the base units link them to fundamental constants of nature, providing a stable and universal basis for measurement.

Besides the seven base units, SI includes two supplementary units, which are dimensionless:

• Radian (rad): Unit for plane angle ($d\theta$). Defined as the ratio of the arc length ($ds$) subtending the angle to the radius ($r$) of the circle ($d\theta = ds/r$).
• Steradian (sr): Unit for solid angle ($d\Omega$). Defined as the ratio of the area ($dA$) intercepted on a spherical surface to the square of its radius ($r$) ($d\Omega = dA/r^2$).

Units for other physical quantities are derived units, formed by combining the base units algebraically (e.g., unit of speed is m/s, unit of force is kg m/s$^2$ which is called Newton, N).

Some derived units have special names (e.g., Newton, Joule, Watt, Volt, Pascal, etc.).

SI also uses standard prefixes (like kilo-, milli-, micro-, nano-) to denote multiples and sub-multiples of units, simplifying the representation of very large or very small values (e.g., 1 km = $10^3$ m, 1 mm = $10^{-3}$ m).

General guidelines exist for the proper use of symbols for physical quantities, units, and prefixes to ensure clarity and consistency in scientific communication.

Measurement Of Length

Measuring length can be done using various methods depending on the scale of the distance.

Direct Methods: Involve using measuring instruments directly:

• Metre Scale: Typically measures lengths from $10^{-3}$ m (millimeter) to $10^2$ m (hundreds of meters).
• Vernier Callipers: Provides greater accuracy, measuring lengths up to $10^{-4}$ m (0.1 mm).
• Screw Gauge / Spherometer: Offer even higher precision for small lengths, measuring up to $10^{-5}$ m (0.01 mm).

For lengths outside these ranges (very large or very small), indirect methods are employed.

Measurement Of Large Distances

Large distances like the distance to a planet or a star cannot be measured directly. The parallax method is a common indirect technique.

Parallax: The apparent shift in the position of an object when viewed from two different points. The distance between the two observation points is called the basis.

To measure the distance ($D$) of a distant object (like a planet $S$) from Earth:

1. Observe the object from two widely separated points ($A$ and $B$) on Earth simultaneously. The distance between $A$ and $B$ is the basis ($b = AB$).
2. Measure the angle ($\theta$) between the two lines of sight ($AS$ and $BS$). This angle is the parallax angle or parallactic angle.
3. Since the object is very far ($D >> b$), $\theta$ is very small. The distance $AB$ can be approximated as an arc of a circle with radius $D$ centered at $S$.
4. The relationship between the basis, distance, and angle (in radians) is given by: $$b = D \theta$$
5. Therefore, the distance is calculated as: $$D = \frac{b}{\theta}$$ (Note: $\theta$ must be in radians).

Once the distance ($D$) to a planet is known, its diameter ($d$) can be determined by measuring its angular size ($\alpha$).

1. Observe the planet through a telescope from one location on Earth.
2. Measure the angle ($\alpha$) subtended by two diametrically opposite points of the planet at the observation point. This is the angular size ($\alpha$).
3. The relationship between diameter, distance, and angular size (in radians) is: $$\alpha = \frac{d}{D}$$
4. The diameter is then calculated as: $$d = \alpha D$$

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Let the distance of the tower C from point A be AC. The man moves from A to B, such that AB = 100 m and AB is perpendicular to AC.

The distant object O is in line with AC when viewed from A. When viewed from B, the line of sight to C is shifted by $\theta = 40^\circ$ relative to the line of sight to O (which is effectively the same direction as AO). This means the angle $\angle ABC = \theta = 40^\circ$.

In the right-angled triangle $\triangle ABC$, we have:

$$\tan \theta = \frac{AB}{AC}$$

We are given $AB = 100$ m and $\theta = 40^\circ$.

$$AC = \frac{AB}{\tan \theta} = \frac{100 \text{ m}}{\tan 40^\circ}$$

Using the value $\tan 40^\circ \approx 0.8391$,

$$AC = \frac{100}{0.8391} \text{ m} \approx 119.17 \text{ m}$$

Rounding to appropriate significant figures (based on 100 m, assuming 2 or 3 sig figs), the distance of the tower C from A is approximately 119 m.

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Estimation Of Very Small Distances: Size Of A Molecule

Measuring extremely small sizes, like those of molecules (ranging from $10^{-8}$ m to $10^{-10}$ m), requires specialized indirect techniques as conventional instruments like screw gauges or even optical microscopes have limitations.

Limitations of Microscopes:

• Optical Microscope: Limited by the wave nature of visible light. Its resolution cannot be better than the wavelength of light (typically $4000 \text{ Å}$ to $7000 \text{ Å}$, where $1 \text{ Å} = 10^{-10} \text{ m}$). It cannot resolve objects smaller than this range.
• Electron Microscope: Uses electron beams, which have much smaller wavelengths than visible light. While offering better resolution (down to $\sim 0.6 \text{ Å}$), it is still limited by the wave nature of electrons.
• Tunnelling Microscopy: Offers resolution better than $1 \text{ Å}$ and can image individual atoms and molecules.

A simple method to estimate the size of a molecule is using oleic acid, which forms a thin, mono-molecular layer on a water surface.

The procedure involves:

1. Preparing a dilute solution of oleic acid in alcohol (e.g., 1 cm$^3$ of oleic acid in 20 cm$^3$ alcohol, then 1 cm$^3$ of this solution diluted to 20 cm$^3$). The final concentration is $\frac{1}{20 \times 20} = \frac{1}{400}$ cm$^3$ of oleic acid per cm$^3$ of solution.
2. Sprinkling lycopodium powder on the surface of water in a large trough. The powder helps to visualize the extent of the oleic acid film.
3. Carefully placing a known number of drops ($n$) of the prepared oleic acid solution onto the water surface.
4. The alcohol evaporates, and the oleic acid spreads to form a thin, roughly circular film of molecular thickness on the water surface, pushing the lycopodium powder outwards.
5. Measuring the diameter of the circular film to calculate its area ($A$).
6. Estimating the volume ($V$) of a single drop of the solution beforehand (e.g., by counting how many drops make 1 cm$^3$).
7. Calculating the total volume of oleic acid in the $n$ drops: $$\text{Volume of oleic acid} = n \times V \times \text{Concentration} = n \times V \times \frac{1}{400} \text{ cm}^3$$
8. Assuming the film is a single layer of molecules, its thickness ($t$) is the volume of oleic acid divided by the area of the film: $$t = \frac{\text{Volume of oleic acid}}{\text{Area of film}} = \frac{nV/400}{A} \text{ cm}$$

This calculated thickness ($t$) provides an estimate for the size or diameter of an oleic acid molecule, which typically comes out to be around $10^{-9}$ m.

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Range Of Lengths

The sizes of objects and distances in the universe vary over an extraordinarily wide spectrum.

The range spans from the size of the smallest known particles (like the diameter of a proton, $\sim 10^{-15}$ m) to the estimated extent of the observable universe ($\sim 10^{26}$ m).

Special units are used for convenience at very small or very large scales:

• fermi (f): $1 \text{ f} = 10^{-15} \text{ m}$ (typical nuclear size scale).
• angstrom (Å): $1 \text{ Å} = 10^{-10} \text{ m}$ (typical atomic size scale).
• astronomical unit (AU): $1 \text{ AU} = 1.496 \times 10^{11} \text{ m}$ (average distance between the Earth and the Sun).
• light year (ly): $1 \text{ ly} = 9.46 \times 10^{15} \text{ m}$ (distance light travels in vacuum in one year).
• parsec (pc): $1 \text{ pc} = 3.08 \times 10^{16} \text{ m}$. Defined as the distance at which the average radius of Earth's orbit (1 AU) subtends a parallax angle of one arc second ($1''$). ($1 \text{ pc} = \frac{1 \text{ AU}}{1'' \text{ in radians}}$).

Measurement Of Mass

Mass is an intrinsic property of matter, representing the amount of substance. It remains constant regardless of temperature, pressure, or location in space.

The SI unit of mass is the kilogram (kg). It is defined based on the fixed numerical value of the Planck constant ($h$).

For dealing with the masses of atoms and molecules, the kilogram is an inconveniently large unit. A more suitable standard unit is the unified atomic mass unit (u).

• $1 \text{ u}$ is defined as exactly one-twelfth ($1/12$) of the mass of an atom of the carbon-12 isotope ($^{12}\text{C}$), including the mass of its electrons.
• $1 \text{ u} \approx 1.66 \times 10^{-27} \text{ kg}$.

Methods for measuring mass depend on the scale:

• Common Balance: Used for measuring the mass of everyday objects (like in shops), comparing an unknown mass with standard masses.
• Gravitational Method: Used for large masses like planets and stars, based on Newton's law of gravitation (related to how they interact gravitationally).
• Mass Spectrograph: Used for measuring the masses of atomic and sub-atomic particles. It works by analyzing the trajectory of charged particles in electric and magnetic fields, where the radius of the path is proportional to the particle's mass.

Range Of Masses

Similar to length, the masses of objects in the universe cover a vast range.

The spectrum extends from the tiny mass of an electron ($\sim 10^{-30}$ kg) to the immense mass of the known observable universe ($\sim 10^{55}$ kg).

This ratio of largest to smallest mass ($\sim 10^{85}$) is the square of the ratio observed for length and time scales ($\sim (10^{41})^2$).

Measurement Of Time

Measuring time intervals requires a device that exhibits a regular, repeating phenomenon – a clock.

The modern standard for time measurement is based on an atomic standard.

• This standard uses the precise, periodic vibrations of radiation corresponding to a specific transition in a Cesium-133 atom.
• This principle is used in caesium clocks (atomic clocks), which serve as national standards for time and frequency.

The SI unit of time, the second (s), is defined based on this atomic standard:

• One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Caesium-133 atom.

Cesium atomic clocks are extremely accurate and provide highly stable timekeeping. For example, they can maintain an uncertainty of $\pm 1 \times 10^{-15}$ parts, meaning they lose or gain no more than about 32 microseconds in one year.

The high accuracy of time measurement has led to the definition of the SI unit of length (the metre) being based on the speed of light and the second.

The time intervals associated with phenomena in the universe also span a tremendous range, from extremely short lifetimes of unstable particles ($\sim 10^{-24}$ s) to the age of the universe ($\sim 10^{18}$ s).

The ratio of the longest to the shortest time intervals is approximately $10^{42}$, which is of the same order of magnitude as the ratio found for length scales ($\sim 10^{40}$). This numerical coincidence across different physical quantities is noteworthy.

Accuracy, Precision Of Instruments And Errors In Measurement

All experimental science and technology rely on measurement. However, every measurement made with any instrument contains some level of uncertainty, referred to as error.

It's important to distinguish between accuracy and precision:

• Accuracy: Describes how close a measured value is to the true or actual value of the quantity. A highly accurate measurement is very near the true value.
• Precision: Refers to the resolution or the limit to which a quantity is measured. It indicates how repeatable the measurements are, i.e., how close multiple measurements are to each other, regardless of their proximity to the true value.

A measurement can be precise but not accurate, or accurate but not precise, or neither, or both.

Due to errors, every measurement is inherently approximate. Errors in measurement can be broadly categorized into:

1. Systematic Errors:
2. Random Errors:

Systematic Errors:

• These errors tend to occur consistently in one direction (either always making the measurement too high or always too low).
• Sources of systematic errors include:
  • Instrumental Errors: Due to imperfect design, faulty calibration (e.g., thermometer reading $104^\circ\text{C}$ at water's boiling point), or zero error (e.g., vernier scale not aligning at zero, worn-off scale ends).
  • Imperfection in Experimental Technique or Procedure: Results from flaws in the experimental setup or method (e.g., measuring body temperature under the armpit gives a lower reading than actual), or external conditions affecting the measurement (temperature changes, air currents).
  • Personal Errors: Arise from the observer's habits, biases, lack of proper setup, or carelessness (e.g., parallax error from consistently viewing a scale from an angle).
• Systematic errors can often be minimised by improving techniques, using better instruments, and correcting for known biases. They can sometimes be estimated and accounted for.

Random Errors:

• These errors occur irregularly and unpredictably, varying in both sign (positive or negative) and magnitude.
• Sources include:
  • Unpredictable fluctuations in experimental conditions (e.g., sudden voltage changes, mechanical vibrations).
  • Unbiased personal errors (slight variations in how an observer takes a reading each time).
• Random errors cannot be eliminated entirely but can be reduced by repeating measurements multiple times and taking the average.

Least Count Error:

• This error is associated with the resolution or the smallest value that can be measured by an instrument (its least count).
• All measurements have an uncertainty related to the least count (e.g., a metre scale with 1 mm divisions has a least count error of $\pm 0.5$ mm or $\pm 1$ mm depending on convention/estimation).
• Least count error contributes to both systematic (if there's a zero error) and random errors.
• It can be reduced by using instruments with higher resolution or by averaging repeated measurements.

Absolute Error, Relative Error And Percentage Error

To quantify errors and represent the uncertainty in a measurement, we use different error metrics.

Suppose a quantity is measured $n$ times, yielding values $a_1, a_2, a_3, ..., a_n$.

• Mean Value: The best estimate of the true value is typically the arithmetic mean of the measurements: $$a_{mean} = \frac{a_1 + a_2 + ... + a_n}{n} = \frac{1}{n}\sum_{i=1}^{n} a_i$$
• Absolute Error ($\Delta a_i$): The difference between an individual measurement and the mean value: $$\Delta a_i = a_i - a_{mean}$$ These individual errors can be positive or negative. The magnitude, $|\Delta a_i|$, is the absolute error.
• Mean Absolute Error ($\Delta a_{mean}$): The average of the magnitudes of the absolute errors from all measurements: $$\Delta a_{mean} = \frac{|\Delta a_1| + |\Delta a_2| + ... + |\Delta a_n|}{n} = \frac{1}{n}\sum_{i=1}^{n} |\Delta a_i|$$ This represents the overall uncertainty in the measurements.

The result of a measurement is conventionally expressed as $a = a_{mean} \pm \Delta a_{mean}$. This indicates that the true value of the quantity is likely to lie within the range $[a_{mean} - \Delta a_{mean}, a_{mean} + \Delta a_{mean}]$.

Other ways to express error include relative error and percentage error:

• Relative Error: The ratio of the mean absolute error to the mean value: $$\text{Relative Error} = \frac{\Delta a_{mean}}{a_{mean}}$$ This is a dimensionless quantity.
• Percentage Error ($\delta a$): The relative error expressed as a percentage: $$\text{Percentage Error} = \frac{\Delta a_{mean}}{a_{mean}} \times 100\%$$

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Combination Of Errors

When calculations involve multiple measured quantities, each with its own error, it's necessary to determine how these individual errors combine to affect the final result. This is known as error propagation.

Rules for error combination:

(a) Error of a Sum or a Difference:

• If $Z = A + B$ or $Z = A - B$, and the measured values are $A \pm \Delta A$ and $B \pm \Delta B$ (where $\Delta A$ and $\Delta B$ are absolute errors).
• The maximum possible absolute error in $Z$ is the sum of the absolute errors: $$\Delta Z = \Delta A + \Delta B$$
• Rule: When quantities are added or subtracted, the absolute error in the result is the sum of the absolute errors in the individual quantities.

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(b) Error of a Product or a Quotient:

• If $Z = AB$ or $Z = A/B$, and the measured values are $A \pm \Delta A$ and $B \pm \Delta B$.
• Assuming $\Delta A$ and $\Delta B$ are small compared to A and B, the maximum relative error in $Z$ is the sum of the relative errors in $A$ and $B$: $$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$
• This rule applies to both multiplication and division.
• Rule: When quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the quantities being combined.

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(c) Error in case of a Measured Quantity Raised to a Power:

• If $Z = A^k$, where $k$ is a constant and the measured value is $A \pm \Delta A$.
• The maximum relative error in $Z$ is the relative error in $A$ multiplied by the power $k$: $$\frac{\Delta Z}{Z} = |k| \frac{\Delta A}{A}$$
• For a more general expression like $Z = \frac{A^p B^q}{C^r}$: $$\frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}$$ (Here we take the magnitudes of the powers and add the relative errors).
• Rule: The relative error in a quantity raised to a power is that power times the relative error in the original quantity. For combined operations involving powers, the maximum relative error in the result is the sum of the relative errors of each factor, multiplied by their respective powers.

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Significant Figures

Since every measurement involves some error or uncertainty, the reported result should reflect the precision of the measurement. Significant figures (or significant digits) are used for this purpose.

The significant figures in a measured value consist of all the digits that are known reliably (certain) plus the first digit that is uncertain or estimated.

For example, if a length is measured as 287.5 cm, the digits 2, 8, and 7 are certain, while 5 is the first uncertain digit. This measurement has four significant figures.

Reporting more digits than are significant is misleading, as it suggests a higher level of precision than was actually achieved.

The number of significant figures in a value indicates the precision of the measurement and is related to the least count of the measuring instrument.

An important rule is that changing the unit of measurement does not change the number of significant figures.

Rules for determining the number of significant figures in a number:

1. All non-zero digits are significant. (e.g., 287.5 has 4 sig figs; 123 has 3 sig figs).
2. Zeros between two non-zero digits are significant. (e.g., 2.008 has 4 sig figs; 101 has 3 sig figs).
3. Leading zeros (zeros to the left of the first non-zero digit) are NOT significant. These zeros only indicate the position of the decimal point. (e.g., $\underline{0}.\underline{00}237$ has 3 sig figs; $\underline{0}.5$ has 1 sig fig).
4. Trailing zeros (zeros at the end of a number):
   • In a number WITHOUT a decimal point, trailing zeros are generally NOT significant. They may just be placeholders for the magnitude. (e.g., 12300 has 3 sig figs; 500 has 1 sig fig).
   • In a number WITH a decimal point, trailing zeros ARE significant. They indicate the precision of the measurement. (e.g., 3.500 has 4 sig figs; 0.06900 has 4 sig figs; 20.0 has 3 sig figs).
5. The zero conventionally placed to the left of the decimal point for a number less than 1 (like 0.1250) is never significant; it's just for readability.

Scientific Notation ($a \times 10^b$) helps resolve ambiguity about trailing zeros in numbers without a decimal point.

• In this form, $a$ is a number between 1 and 10, and $b$ is an integer power of 10.
• All digits shown in the number 'a' are considered significant.
• The power of 10 ($10^b$) is irrelevant to the number of significant figures.
• Example: 12300 can be written as $1.23 \times 10^4$ (3 sig figs), $1.230 \times 10^4$ (4 sig figs), or $1.2300 \times 10^4$ (5 sig figs) depending on its known precision.
• The exponent $b$ in scientific notation is called the order of magnitude of the quantity.

Exact Numbers: Quantities that are counted or defined (like the '2' in $2\pi r$) are considered to have an infinite number of significant figures. This means they do not limit the number of significant figures in a calculation result.

Rules For Arithmetic Operations With Significant Figures

When performing calculations with measured values, the result's precision is limited by the least precise input value. The rules for handling significant figures in arithmetic operations are:

(1) In Multiplication or Division:

• The final result should have the same number of significant figures as the input number with the least number of significant figures.
• Example: Calculating density = mass / volume. If mass = 4.237 g (4 sig figs) and volume = 2.51 cm$^3$ (3 sig figs), the result should have 3 significant figures. $4.237 / 2.51 \approx 1.688$. Rounded to 3 sig figs, density = 1.69 g/cm$^3$.
• Example: Calculating light year distance. If speed of light = $3 \times 10^8$ m/s (1 sig fig) and seconds in a year = $3.156 \times 10^7$ s (4 sig figs), the result should have 1 significant figure. $3 \times 10^8 \times 3.156 \times 10^7 \approx 9.468 \times 10^{15}$. Rounded to 1 sig fig, light year $\approx 9 \times 10^{15}$ m (though in practice, usually more sig figs are used for constants).

(2) In Addition or Subtraction:

• The final result should retain the same number of decimal places as the input number with the least number of decimal places.
• Example: Summing masses 436.32 g (2 decimal places), 227.2 g (1 decimal place), and 0.301 g (3 decimal places). The least precise is 227.2 g with 1 decimal place. $$436.32$$ $$227.2\underline{0}$$ $$\underline{0.301}$$ $$663.821$$ The sum is 663.821 g. Rounding to 1 decimal place, the result is 663.8 g.
• Example: Difference in length 0.307 m (3 decimal places) - 0.304 m (3 decimal places) = 0.003 m (3 decimal places). This result has 1 significant figure, but the rule is based on decimal places for addition/subtraction.

Rounding Off The Uncertain Digits

When a calculation produces more digits than are significant, the result must be rounded off to the appropriate number of significant figures.

Standard rules for rounding:

1. If the digit to be dropped is greater than 5, increase the preceding digit by one. (e.g., 2.746 rounded to 3 sig figs becomes 2.75).
2. If the digit to be dropped is less than 5, leave the preceding digit unchanged. (e.g., 2.743 rounded to 3 sig figs becomes 2.74).
3. If the digit to be dropped is exactly 5:
   • If the preceding digit is even, leave it unchanged. (e.g., 2.745 rounded to 3 sig figs becomes 2.74).
   • If the preceding digit is odd, increase it by one. (e.g., 2.735 rounded to 3 sig figs becomes 2.74).

In multi-step calculations, it is good practice to keep at least one extra digit beyond the significant figures in intermediate steps to avoid accumulation of rounding errors, and only round the final result to the correct number of significant figures.

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Rules For Determining The Uncertainty In The Results Of Arithmatic Calculations

Beyond simply counting significant figures, we can also determine the actual uncertainty (error) in the result of calculations involving measured quantities with known uncertainties.

These rules are essentially a re-application or illustration of the error combination rules (Section 2.6.2) in the context of reporting the final result with significant figures and uncertainty.

(1) Uncertainty in Product (Example):

• If length $l = 16.2 \pm 0.1$ cm (16.2 cm is mean value, 0.1 cm is absolute error) and breadth $b = 10.1 \pm 0.1$ cm.
• $l$ has 3 significant figures; $\Delta l = 0.1$ cm. Relative error in $l = \frac{0.1}{16.2} \times 100\% \approx 0.6\%$.
• $b$ has 3 significant figures; $\Delta b = 0.1$ cm. Relative error in $b = \frac{0.1}{10.1} \times 100\% \approx 1.0\%$.
• Area $A = l \times b = 16.2 \text{ cm} \times 10.1 \text{ cm} = 163.62 \text{ cm}^2$.
• Relative error in Area $\frac{\Delta A}{A} = \frac{\Delta l}{l} + \frac{\Delta b}{b}$.
• Percentage error in Area = Percentage error in $l$ + Percentage error in $b \approx 0.6\% + 1.0\% = 1.6\%$.
• Absolute error in Area $\Delta A = A \times (\text{Relative error in A}) = 163.62 \times (1.6/100) \approx 163.62 \times 0.016 \approx 2.618 \text{ cm}^2$.
• Rounding the absolute error to one or two significant figures, say 3 cm$^2$.
• The area is reported as $163.62 \pm 2.618 \text{ cm}^2$. Given the uncertainty $\pm 2.618$, the digit '3' in 163 is uncertain. The result should be rounded to reflect this. 163.62 rounded to the nearest integer (because the error is around 3) is 164.
• The area is reported as $(164 \pm 3) \text{ cm}^2$. This shows the mean value rounded appropriately and the absolute uncertainty.

(2) Uncertainty in Subtraction (Impact on Sig Figs):

• If subtracting values with the same number of significant figures, the result may have fewer significant figures.
• Example: $12.9$ g (3 sig figs, 1 decimal place) - $7.06$ g (3 sig figs, 2 decimal places).
• Using the rule for addition/subtraction (least decimal places), the result should have 1 decimal place.
$$12.9$$ $$-\underline{7.06}$$ $$5.84$$

Rounding 5.84 to 1 decimal place gives 5.8 g. This result (5.8) has only 2 significant figures, less than the original numbers.

(3) Relative Error Dependence:

• The relative error for a number with $n$ significant figures depends not just on $n$, but also on the value of the number itself.
• Example:
  • Measurement 1: 1.02 g. Absolute error $\pm 0.01$ g (uncertainty in the last digit). Relative error = $\frac{0.01}{1.02} \times 100\% \approx 0.98\% \approx 1\%$.
  • Measurement 2: 9.89 g. Absolute error $\pm 0.01$ g. Relative error = $\frac{0.01}{9.89} \times 100\% \approx 0.101\% \approx 0.1\%$.
• Both measurements have 3 significant figures and the same absolute error magnitude ($\pm 0.01$), but their relative errors are different.

General Advice for Complex Calculations:

• In intermediate steps of multi-step calculations, carry at least one more significant figure than required for the final result (based on the least precise input). This prevents rounding errors from accumulating and affecting the final precision.

Dimensions Of Physical Quantities

The fundamental nature of a physical quantity, independent of the specific system of units used for measurement, is described by its dimensions.

All physical quantities can be expressed in terms of a combination of the seven fundamental or base quantities defined in SI.

These seven base quantities are considered the seven dimensions of the physical world. They are denoted using square brackets [ ]:

• Length: [L]
• Mass: [M]
• Time: [T]
• Electric Current: [A]
• Thermodynamic Temperature: [K]
• Amount of Substance: [mol]
• Luminous Intensity: [cd]

The dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent that quantity.

For example, in mechanics, quantities can often be expressed using only the dimensions of length, mass, and time ([L], [M], [T]).

• Volume: Defined by Length $\times$ Breadth $\times$ Height. Dimensions are $[L] \times [L] \times [L] = [L]^3$. In terms of all base dimensions: $[M^0 L^3 T^0 A^0 K^0 mol^0 cd^0]$. We typically only show the non-zero dimensions.
• Force: Defined by Mass $\times$ Acceleration. Acceleration is (Length / Time$^2$). So, Force has dimensions $[M] \times \frac{[L]}{[T]^2} = [MLT^{-2}]$. This means force has one dimension in mass, one dimension in length, and minus two dimensions in time.
• Velocity or Speed: Defined by Distance / Time. Dimensions are $\frac{[L]}{[T]} = [LT^{-1}]$.

When considering dimensions, we focus on the fundamental nature or 'quality' of the physical quantity, not its magnitude or whether it is a vector or scalar. Quantities of the same type (like speed, velocity, initial velocity) have the same dimensions.

Dimensional Formulae And Dimensional Equations

A dimensional formula is an expression that shows which base quantities represent the dimensions of a physical quantity and to what powers they are raised.

• Dimensional formula of Volume: $[M^0 L^3 T^0]$
• Dimensional formula of Speed/Velocity: $[M^0 L T^{-1}]$
• Dimensional formula of Acceleration: $[M^0 L T^{-2}]$
• Dimensional formula of Mass Density: $[M L^{-3} T^0]$

A dimensional equation is formed by equating a physical quantity symbol (placed within square brackets) to its dimensional formula.

• Dimensional equation of Volume: $[V] = [M^0 L^3 T^0]$
• Dimensional equation of Speed/Velocity: $[v] = [M^0 L T^{-1}]$
• Dimensional equation of Force: $[F] = [M L T^{-2}]$
• Dimensional equation of Mass Density: $[\rho] = [M L^{-3} T^0]$

Dimensional formulae and equations can be derived from the defining equations or relationships between physical quantities.

Dimensional Analysis And Its Applications

Understanding the dimensions of physical quantities is crucial because it leads to a powerful tool called dimensional analysis.

A fundamental principle of dimensional analysis is the Principle of Homogeneity of Dimensions: Physical quantities can only be added to or subtracted from other quantities if they have the same dimensions. Similarly, the terms on both sides of a valid equation must have the same dimensions.

Just as units can be treated as algebraic symbols (cancelled in numerator/denominator), dimensions can also be manipulated algebraically.

Dimensional analysis has several key applications:

1. Checking the dimensional consistency (correctness) of equations.
2. Deducing possible relationships between physical quantities (within limitations).

Checking The Dimensional Consistency Of Equations

To check if an equation is dimensionally consistent (homogeneous), verify that the dimensions of every term on both sides of the equation are the same.

If the dimensions of terms in an equation are not the same, the equation is definitely wrong.

If the dimensions are the same, the equation is dimensionally consistent. However, a dimensionally consistent equation is not necessarily correct. Dimensional analysis cannot verify dimensionless constants (like 1/2, $2\pi$) or the correctness of trigonometric, logarithmic, or exponential functions.

Important considerations:

• Arguments of dimensionless functions (like $\sin(\theta)$, $\log(x)$, $e^y$) must be dimensionless. For $\sin(2\pi t/T)$, the argument $2\pi t/T$ must be dimensionless. Dimension of $t$ is [T], dimension of $T$ is [T], $2\pi$ is dimensionless. $[T]/[T]$ is dimensionless.
• Pure numbers or ratios of similar physical quantities (like angle = arc length/radius, refractive index = speed/speed) are dimensionless.

Example: Check the dimensional consistency of the equation of motion: $x = x_0 + v_0 t + \frac{1}{2}at^2$, where $x, x_0$ are positions (length), $v_0$ is initial velocity, $a$ is acceleration, and $t$ is time.

• Dimension of the left side term $[x]$: $[L]$
• Dimension of the first term on the right side $[x_0]$: $[L]$
• Dimension of the second term on the right side $[v_0 t]$: $[LT^{-1}] \times [T] = [L]$
• Dimension of the third term on the right side $[\frac{1}{2}at^2]$: The constant $\frac{1}{2}$ is dimensionless. Dimension of $a$ is $[LT^{-2}]$, dimension of $t^2$ is $[T^2]$. So, $[LT^{-2}] \times [T^2] = [L]$.

Since every term has the dimension of length [L], the equation is dimensionally consistent.

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Deducing Relation Among The Physical Quantities

Dimensional analysis can sometimes help in deducing a possible relationship between a physical quantity and other quantities it depends on. This method works best when the dependence is of a product type and involves a limited number of variables (typically up to three, in simple cases).

The procedure is to assume the physical quantity is proportional to a product of the other quantities raised to unknown powers, and then use dimensional homogeneity to solve for the exponents.

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Limitations of Dimensional Analysis:

• Cannot determine dimensionless constants ($k$).
• Cannot determine the exact mathematical form of a relationship if it involves sums, differences, logarithmic, exponential, or trigonometric functions (as these don't follow the simple product rule assumed).
• Cannot distinguish between physical quantities that have the same dimensions (e.g., work and torque both have dimensions $[ML^2 T^{-2}]$).
• Can only deduce relations for quantities that depend on a limited number of other quantities in a simple product form.

Despite its limitations, dimensional analysis is a powerful and useful tool for checking consistency and gaining insight into the possible forms of physical relationships.

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