Units and Measurements Comprehensive Study Guide

Fundamental Concepts of Units and Measurements

A unit is defined as an internationally accepted reference standard employed to measure a physical quantity. In the study of physics, quantities are distinguished based on their dependency. Fundamental quantities are physical quantities that remain independent of each other, such as length, mass, and time. The specific units chosen for the measurement of these fundamental quantities are referred to as fundamental or base units. Conversely, derived units are those chosen for the measurement of derived quantities, which are calculated from base quantities. A complete set of units, incorporating both fundamental and derived units, is known as a system of units.

International Systems of Units and SI Standards

Historically, three systems of units were used extensively across various countries: the CGS, FPS, and MKS systems. In the CGS system, the units of length, mass, and time are the centimetre, gram, and second, respectively. The FPS system utilizes the foot, pound, and second. The MKS system employs the metre, kilogram, and second. Modern science relies on the SI (Système International d'Unités) system, which provides several advantages. It is a rationalized system, meaning only one unit is used for any single physical quantity, and it is a decimal system, making conversions within the system simple and convenient.

The SI system identifies seven fundamental units and two supplementary units. The seven base quantities and their corresponding units and symbols are: mass measured in kilogram ($kg$), length in metre ($m$), time in second ($s$), temperature in kelvin ($K$), electric current in ampere ($A$), luminous intensity in candela ($cd$), and the quantity of matter in mole ($mol$). The two supplementary physical quantities are the plane angle, measured in radian ($rad$), and the solid angle, measured in steradian ($sr$).

Standard SI Prefixes

To represent scales of magnitude, the SI system utilizes specific prefixes for powers of ten. For sub-multiples of units, the prefixes include: milli for $10^{-3}$ ($m$), micro for $10^{-6}$ ($\mu$), nano for $10^{-9}$ ($n$), and pico for $10^{-12}$ ($p$). For multiples of units, the prefixes include: kilo for $10^3$ ($k$), mega for $10^6$ ($M$), giga for $10^9$ ($G$), and tera for $10^{12}$ ($T$). These allow for the efficient communication of extremely large or small physical measurements.

Methods for Measuring Length and Distance

Length measurement can be categorized into direct and indirect methods. Direct methods involve the use of instruments such as the metre scale, vernier callipers, and screw gauges. Indirect methods, such as the parallax method, are required to measure the distance of a planet or star from the earth. Parallax is defined as the apparent change in the position of an object with respect to a reference point on a wall when viewed from the left and the right eye. The distance between the two points of observation is called the basis; for example, the distance between a person's eyes. The angle between the two directions along which the object is viewed is known as the parallax angle or parallactic angle ($\theta$).

To measure the distance $D$ of a faraway planet $S$, it is observed from two different positions or observatories, $A$ and $B$, on the earth, separated by a distance $b$. Using the principle that the length of an arc is equal to the product of the radius of the arc and the angle subtended at the centre, we derive the formula $b = D \times \theta$. By knowing the basis $b$ and measuring the angle $\theta$, the distance $D$ can be calculated. To determine the size or diameter $d$ of a planet, two diametrically opposite points $A$ and $B$ on the planet are observed from a single point $O$ on earth. The angular diameter $\alpha$ is measured using a telescope, and the diameter is calculated as $d = \alpha \times D$. Regarding angular unit conversions, $1^{\circ}$ is equivalent to $1.745 \times 10^{-2}\,rad$, $1'$ (one arc minute) equals $2.91 \times 10^{-4}\,rad$, and $1''$ (one arc second) equals $4.847 \times 10^{-6}\,rad$.

Estimation of Molecular Size via the Oleic Acid Method

Small measurements, such as the size of an oleic acid molecule, require specific dilution techniques. Initially, $1\,cm^3$ of oleic acid is dissolved in alcohol to make a solution of $20\,cm^3$. Then, $1\,cm^3$ of this solution is diluted further in alcohol to reach a final volume of $20\,cm^3$. The resulting concentration is $\frac{1}{20 \times 20}\,cm^3$ of oleic acid per $cm^3$ of solution. Lycopodium powder is sprinkled on the water surface in a large trough, and $n$ drops of the solution, each of volume $V$, are added. The volume of oleic acid in the solution is given by $nV \times \frac{1}{20 \times 20}\,cm^3$. The solution spreads into a thin film of area $A$ and thickness $t$. As the alcohol evaporates, the volume of the remaining thin film is defined as $A \times t$. By equating the volumes, the thickness is calculated as $t = \frac{nV}{20 \times 20 \times A}$. This thickness represents the size of the oleic acid molecules.

Ranges of Length, Mass, and Time

The universe exhibits a vast range of lengths. The nuclear size is on the order of $10^{-14}\,m$, while the observable universe spans approximately $10^{26}\,m$. Special units for short distances include the Fermi ($1\,fm = 10^{-15}\,m$) and the Angstrom ($1\,\text{\AA} = 10^{-10}\,m$). For large lengths, units include the Astronomical Unit ($1\,AU = 1.496 \times 10^{11}\,m$, representing the average distance of the sun from the earth), the Light year ($1\,ly = 9.46 \times 10^{15}\,m$, the distance light travels in one year at $3 \times 10^8\,m/s$), and the Parsec ($3.08 \times 10^{16}\,m$). The parsec is the largest unit of distance and is defined as the distance at which the average radius of the earth's orbit subtends an angle of $1\,\text{arc second}$. The ratio of the longest to shortest lengths in the universe is approximately $10^{41}$.

Mass measurements vary from atoms to celestial bodies. The mass of atoms is expressed in the unified atomic mass unit ($u$), defined as $\frac{1}{12}$ the mass of one atom of the $C-12$ isotope ($1\,u = 1.66 \times 10^{-27}\,kg$). The masses of planets and stars are measured via gravitational methods, while atomic and subatomic particle masses are measured using a mass spectrograph. This device operates on the principle that the radius of the trajectory of a charged particle moving through uniform electric and magnetic fields is proportional to its mass. The range of mass extends from an electron ($10^{-30}\,kg$) to the universe ($10^{55}\,kg$), a ratio of $10^{85}$.

Time is measured accurately using the cesium atomic clock, which is based on the periodic vibrations of the cesium atom. The ratio of the longest time interval (the age of the universe) to the shortest (time associated with subatomic events) is approximately $10^{41}$.

Accuracy, Precision, and Measurement Errors

Accuracy indicates how close a measured value is to the true value of a quantity, whereas precision refers to the resolution or limit of the instrument used. For instance, if a true length is $3.678\,cm$, a measurement of $3.5\,cm$ with a least count of $0.1\,cm$ is more accurate, while a measurement of $3.38\,cm$ with a least count of $0.01\,cm$ is more precise. Error is the uncertainty in measurement. Errors are classified as systematic or random. Systematic errors tend to be in one direction (positive or negative) and include instrumental errors (due to poor design or calibration, like a thermometer reading $104^{\circ}C$ for boiling water), imperfections in experimental techniques (like reading armpit temperature incorrectly), and personal errors (like individual parallax bias). Systematic error is minimized by improving techniques and using better calibrated instruments. Least count error is the error linked to the instrument's resolution and is reduced by using high-precision instruments.

Mathematical Treatment of Errors and Propagation

When multiple trials are taken, the arithmetic mean ($a_{mean} = \frac{\sum a_i}{n}$) serves as the true value. The absolute error is the difference magnitude between an individual measurement and the true value ($\Delta a_i = |a_i - a_{mean}|$). Mean absolute error is the arithmetic mean of all absolute errors ($\Delta a_{mean}$). Relative error is the ratio $\Delta a_{mean} / a_{mean}$, and percentage error is the relative error multiplied by $100$.

Error propagation rules define how errors combine in calculations: when two quantities are added or subtracted, the absolute error in the result is the sum of the absolute errors of the individual quantities ($\Delta Z = \Delta A + \Delta B$). When quantities are multiplied or divided, the relative error of the result is the sum of the relative errors of the multipliers ($\Delta Z / Z = \Delta A / A + \Delta B / B$). If a quantity is raised to a power $k$, the relative error in the result is $k$ times the relative error of the individual quantity.

Significant Figures and Dimensional Analysis

Significant figures consist of reliable digits plus the first uncertain digit. They do not depend on the unit system used. Standard rules state that all non-zero digits are significant; zeros between non-zeros are significant; leading zeros in decimals less than one are not significant; trailing zeros without a decimal point are not significant; and trailing zeros with a decimal are significant. Scientific notation ($a \times 10^b$) is used where the number of digits in $a$ indicates significance. Rounding rules state that if the digit to be dropped is more than $5$, the preceding digit increases by $1$; if less than $5$, it remains unchanged. If the digit is exactly $5$, the preceding digit increases by $1$ if it is odd and remains the same if even.

Dimensions are the powers to which base quantities are raised to represent a specific quantity. A dimensional formula shows this relationship, while a dimensional equation relates a physical quantity to fundamental ones. Some quantities like impulse and momentum share the same dimensions ($[M^1L^1T^{-1}]$). Others, like the plane angle, have units but no dimensions, while quantities like refractive index or Reynold's number have neither. The principle of homogeneity states that the dimensions of all terms in a physical equation must be the same. This principle is used to check equation correctness, convert units, and derive relations. However, it cannot determine proportionality constants, handle trigonometric or logarithmic functions, or work when a quantity depends on more than the three base units of length, mass, and time.

Questions & Discussion

What is the ratio of longest and shortest time intervals in the universe? The ratio is calculated as $10^{17}\,s / 10^{-24}\,s = 10^{41}$.

Give an example of a constant with no unit. Reynold number is an example of a unitless constant.

How do you convert Newtons to Dynes? Force has dimensions $[MLT^{-2}]$. Since $1\,N = 1\,kg \cdot (1\,m) \cdot (1\,s)^{-2}$, and $1\,kg = 10^3\,g$ with $1\,m = 10^2\,cm$, the conversion yields $10^3 \times 10^2 = 10^5\,dyne$.

How do you convert Joules to Ergs? Energy has dimensions $[ML^2T^{-2}]$. Converting $1\,kg$ to $10^3\,g$ and $1\,m$ to $10^2\,cm$ results in $10^3 \times (10^2)^2 = 10^7\,erg$.

How was the distance to the moon computed based on the transcript data? Using the parallax method with an angle $\theta = 1^{\circ}54' = 3.32 \times 10^{-2}\,rad$ and a diameter of Earth (basis) $b = 1.276 \times 10^7\,m$, the distance $D = b / \theta$ results in $3.84 \times 10^8\,m$.