Significant Figures

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### Introduction to Measurement - **Measurement:** The process of comparing an unknown physical quantity with a known standard quantity of the same kind. - **Physical Quantity:** A quantity that can be measured (e.g., length, mass, time, force). - **Unit:** A standard reference chosen to measure a physical quantity. ### Types of Physical Quantities 1. **Fundamental (Base) Quantities:** Independent of other physical quantities. * **Examples:** Length, Mass, Time, Electric Current, Thermodynamic Temperature, Amount of Substance, Luminous Intensity. 2. **Derived Quantities:** Quantities that can be expressed in terms of fundamental quantities. * **Examples:** Speed (Length/Time), Force (Mass $\times$ Length/Time$^2$), Density (Mass/Volume). ### Systems of Units - **CGS System:** Centimeter, Gram, Second. - **FPS System:** Foot, Pound, Second. - **MKS System:** Meter, Kilogram, Second. - **SI System (Système International d'Unités):** The internationally accepted system of units, based on 7 fundamental units and 2 supplementary units. It is a rationalized MKS system. ### SI Fundamental Units | Physical Quantity | SI Unit | Symbol | | :----------------------- | :------------ | :----- | | Length | meter | m | | Mass | kilogram | kg | | Time | second | s | | Electric Current | ampere | A | | Thermodynamic Temperature | kelvin | K | | Amount of Substance | mole | mol | | Luminous Intensity | candela | cd | #### Supplementary Units - **Plane Angle:** radian (rad) - **Solid Angle:** steradian (sr) ### SI Derived Units (Examples) - **Area:** $m^2$ - **Volume:** $m^3$ - **Density:** $kg/m^3$ - **Speed/Velocity:** $m/s$ - **Acceleration:** $m/s^2$ - **Force:** Newton (N) = $kg \cdot m/s^2$ - **Pressure:** Pascal (Pa) = $N/m^2$ - **Work/Energy:** Joule (J) = $N \cdot m$ - **Power:** Watt (W) = $J/s$ - **Frequency:** Hertz (Hz) = $s^{-1}$ ### Prefixes for SI Units | Prefix | Symbol | Factor | | :----- | :----- | :----------- | | tera | T | $10^{12}$ | | giga | G | $10^9$ | | mega | M | $10^6$ | | kilo | k | $10^3$ | | hecto | h | $10^2$ | | deca | da | $10^1$ | | deci | d | $10^{-1}$ | | centi | c | $10^{-2}$ | | milli | m | $10^{-3}$ | | micro | $\mu$ | $10^{-6}$ | | nano | n | $10^{-9}$ | | pico | p | $10^{-12}$ | | femto | f | $10^{-15}$ | ### Measurement of Length - **Direct Methods:** * **Meter Scale:** Least count (LC) = 1 mm or 0.1 cm. * **Vernier Calipers:** LC = 0.1 mm or 0.01 cm. Used for internal/external diameters, depth. * **Screw Gauge/Spherometer:** LC = 0.01 mm or 0.001 cm. Used for very small lengths, wire diameter, thickness of a sheet. - **Indirect Methods (for large distances):** * **Parallax Method:** Used for measuring distances to planets or stars. * $D = \frac{b}{\theta}$, where $D$ is distance, $b$ is basis, $\theta$ is parallax angle (in radians). ### Measurement of Mass - **Common Balance:** For ordinary masses. - **Inertial Balance:** For measuring inertial mass in space. - **Mass Spectrograph:** For atomic/molecular masses. - **Atomic Mass Unit (amu):** $1 \text{ amu} = 1.66056 \times 10^{-27} \text{ kg}$. ### Measurement of Time - **Atomic Clock:** Based on periodic vibrations of Cesium-133 atoms. Highly accurate and precise. - **Standard:** 1 second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the Cesium-133 atom. ### Accuracy, Precision & Errors in Measurement - **Accuracy:** How close a measurement is to the true value. - **Precision:** How close repeated measurements are to each other (resolution of the instrument). * A precise measurement may not be accurate. * A high-accuracy measurement is usually precise. #### Errors in Measurement 1. **Systematic Errors:** Tend to be in one direction (positive or negative). * **Instrumental errors:** Due to faulty calibration (e.g., zero error in scale). * **Imperfect experimental technique:** Due to faulty procedure (e.g., heat loss in calorimetry). * **Personal errors:** Due to individual bias (e.g., parallax error). * **Minimization:** Can be minimized by improving experimental techniques, using better instruments, and removing personal bias. 2. **Random Errors:** Occur irregularly and are random in magnitude and direction. * **Causes:** Unpredictable fluctuations in experimental conditions, personal judgment by observer. * **Minimization:** Can be minimized by taking a large number of observations and calculating the arithmetic mean. 3. **Gross Errors:** Due to carelessness of the observer (e.g., wrong reading, improper recording). * **Minimization:** Careful observation, reading, and recording. #### Estimation of Errors - **Absolute Error ($\Delta a$):** Magnitude of the difference between the true value and the individual measurement. * True value is often taken as the arithmetic mean of multiple measurements: $a_{mean} = \frac{a_1 + a_2 + ... + a_n}{n}$. * $\Delta a_i = |a_{mean} - a_i|$. * Mean absolute error: $\Delta a_{mean} = \frac{|\Delta a_1| + |\Delta a_2| + ... + |\Delta a_n|}{n}$. * Measurement is expressed as $a = a_{mean} \pm \Delta a_{mean}$. - **Relative Error:** $\delta a = \frac{\Delta a_{mean}}{a_{mean}}$. - **Percentage Error:** $\delta a \times 100\% = \frac{\Delta a_{mean}}{a_{mean}} \times 100\%$. #### Combination of Errors - **For Sum or Difference ($Z = A \pm B$):** Absolute errors add up. * $\Delta Z = \Delta A + \Delta B$. - **For Product or Quotient ($Z = A \times B$ or $Z = A/B$):** Relative errors add up. * $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$. - **For Quantity Raised to a Power ($Z = A^p B^q / C^r$):** * $\frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}$. ### Dimensional Analysis - **Dimensions:** The powers to which the fundamental units are raised to represent a derived unit. Represented by $[M]$, $[L]$, $[T]$, $[A]$, $[K]$, $[mol]$, $[cd]$. * **Example:** Dimension of Speed = $[L][T]^{-1}$. - **Dimensional Formula:** An expression showing the powers to which base units are raised. - **Dimensional Equation:** Equation obtained by equating a physical quantity with its dimensional formula. #### Principle of Homogeneity of Dimensions - States that the dimensions of all the terms on both sides of a physical equation must be the same. - Only physical quantities with the same dimensions can be added or subtracted. #### Applications of Dimensional Analysis 1. **Checking the consistency (correctness) of a physical equation:** * If dimensions on LHS $\ne$ dimensions on RHS, the equation is incorrect. * **Example:** $v = u + at$ * $[L][T]^{-1} = [L][T]^{-1} + [L][T]^{-2}[T] = [L][T]^{-1} + [L][T]^{-1}$. Consistent. 2. **Deriving relations among physical quantities:** * Assume a relation of the form $X = k A^a B^b C^c$, then equate powers of dimensions. * **Example:** Period of simple pendulum $T \propto m^a L^b g^c$. * $[T] = [M]^a [L]^b ([L][T]^{-2})^c = [M]^a [L]^{b+c} [T]^{-2c}$. * Comparing powers: $a=0$, $b+c=0$, $-2c=1 \Rightarrow c=-1/2$. * So, $b=1/2$. Thus $T = k \sqrt{L/g}$. 3. **Converting units from one system to another:** * $n_1[U_1] = n_2[U_2]$, where $[U] = [M^a L^b T^c]$. * $n_2 = n_1 \left( \frac{M_1}{M_2} \right)^a \left( \frac{L_1}{L_2} \right)^b \left( \frac{T_1}{T_2} \right)^c$. * **Example:** Convert 1 Joule to ergs ($1 \text{ J} = 1 \text{ kg} \cdot m^2 \cdot s^{-2}$, $1 \text{ erg} = 1 \text{ g} \cdot cm^2 \cdot s^{-2}$). * $n_2 = 1 \left( \frac{kg}{g} \right)^1 \left( \frac{m}{cm} \right)^2 \left( \frac{s}{s} \right)^{-2} = 1 \left( \frac{1000g}{g} \right)^1 \left( \frac{100cm}{cm} \right)^2 (1)^{-2} = 10^3 \times (10^2)^2 = 10^3 \times 10^4 = 10^7$. * So, $1 \text{ Joule} = 10^7 \text{ ergs}$. #### Limitations of Dimensional Analysis - Cannot determine dimensionless constants (like $k$ in $T = k \sqrt{L/g}$). - Cannot be used for equations involving trigonometric, exponential, or logarithmic functions. - Cannot be used if a physical quantity depends on more than three fundamental quantities (e.g., if it depends on mass, length, time, and temperature). - If an equation contains terms added or subtracted, it cannot distinguish them (e.g., $s = ut + \frac{1}{2}at^2$ can only verify $[L] = [L] + [L]$).