Physics and Measurement Study Notes: Standards of Measurement and Analysis

Introduction to Physics and Measurement

• Foundational Objectives of Physics: Physics is an experimental science based on quantitative measurements. Its main goals are to identify a limited number of fundamental laws governing natural phenomena and to use these laws to develop theories capable of predicting results of future experiments.

• The Language of Mathematics: Mathematics provides the bridge between theory and experiment. Laws are expressed mathematically to enable precise predictions.

• Theory Evolution and Modification: When discrepancies exist between theoretical predictions and experimental results, theories must be modified. A theory may be valid only under specific conditions.

  • Newton’s Laws of Motion: Accurately describe motion at normal speeds but fail at speeds near the speed of light.

  • Einstein’s Special Theory of Relativity: Correctly describes motion at all speeds, including those approaching light speed, reducing to Newton’s laws at low speeds. It is a more general theory.

• Eras of Physics:

  • Classical Physics: Includes mechanics, thermodynamics, optics, and electromagnetism developed before 1900. Newton was a primary contributor, also originating calculus.

  • Modern Physics: Began near the end of the 19th century. Key developments include relativity and quantum mechanics.

  • Relativity: Modifies concepts of space, time, and energy; establishes the speed of light as an upper limit; relates mass and energy.

  • Quantum Mechanics: Formulated to describe physical phenomena at the atomic level.

• Technological Applications: Modern physics research has led to unmanned planetary exploration, nanotechnology, microcircuitry, high-speed computers, medical imaging, and genetic engineering.

Standards of Length, Mass, and Time

• Fundamental Quantities: In mechanics, the three fundamental quantities are length, mass, and time. All other quantities are derived from these.

• Requirements for Standards: To ensure reproducibility, a standard must be readily accessible, yield the same result regardless of location (be universal), and remain constant over time.

• SI (Systéme International) Units: Established in 1960.

  • Length: Meter ($m$)

  • Mass: Kilogram ($kg$)

  • Time: Second ($s$)

  • Other Fundamental Standards: Kelvin ($K$) for temperature, Ampere ($A$) for electric current, Candela ($cd$) for luminous intensity, and Mole ($mol$) for the amount of substance.

Standard of Length

• Definition: The distance between two points in space.

• Historical Standards:

  • Yard (1120): Distance from the tip of the King of England’s nose to the end of his outstretched arm.

  • Foot (French): Length of King Louis XIV’s foot.

  • Meter (1799): One ten-millionth ($1/10\,000\,000$) of the distance from the equator to the North Pole along a longitudinal line through Paris. This was not universal as it was Earth-based.

  • Platinum–Iridium Meter (Until 1960): Distance between two lines on a specific bar kept in France.

  • Krypton-86 Meter (1960s-1970s): Defined as $1\,650\,763.73$ wavelengths of orange-red light emitted from a krypton-86 lamp.

• Current SI Definition (1983): The distance traveled by light in a vacuum during a time interval of $\frac{1}{299\,792\,458}$ second. This effectively sets the speed of light in vacuum at exactly $299\,792\,458\,m/s$.

• Approximate Values of Measured Lengths:

  • Distance from Earth to the most remote known quasar: $1.4 \times 10^{26}\,m$

  • One light-year: $9.46 \times 10^{15}\,m$

  • Mean orbit radius of Earth around Sun: $1.50 \times 10^{11}\,m$

  • Mean distance from Earth to Moon: $3.84 \times 10^{8}\,m$

  • Mean radius of the Earth: $6.37 \times 10^{6}\,m$

  • Length of a football field: $9.1 \times 10^{1}\,m$

  • Size of cells: $10^{-5}\,m$

  • Diameter of a proton: $10^{-15}\,m$

Standard of Mass

• Definition: The SI fundamental unit is the kilogram ($kg$).

• Physical Standard: Defined by the mass of a specific platinum–iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sévres, France (established in 1887). Platinum–iridium is used due to its extreme stability.

• Approximate Masses:

  • Observable Universe: $\sim 10^{52}\,kg$

  • Milky Way Galaxy: $\sim 10^{42}\,kg$

  • Sun: $1.99 \times 10^{30}\,kg$

  • Earth: $5.98 \times 10^{24}\,kg$

  • Human: $\sim 10^{2}\,kg$

  • Hydrogen atom: $1.673 \times 10^{-27}\,kg$

  • Electron: $9.11 \times 10^{-31}\,kg$

Standard of Time

• Historical Definition (Pre-1967): Defined by the mean solar day. One second was defined as $(\frac{1}{60})(\frac{1}{60})(\frac{1}{24})$ of a mean solar day. This was not universal as it relied on Earth’s rotation.

• Current SI Definition (1967): One second ($s$) is defined as $9\,192\,631\,770$ times the period of vibration of radiation from the cesium-133 atom. These measurements are taken using atomic clocks, which are accurate to within 1 second every 20 million years.

• Approximate Time Intervals:

  • Age of the Universe: $4 \times 10^{17}\,s$

  • Age of the Earth: $1.3 \times 10^{17}\,s$

  • One year: $3.2 \times 10^{7}\,s$

  • One day: $8.6 \times 10^{4}\,s$

  • Time interval for light to cross a proton: $\sim 10^{-24}\,s$

Units and Derived Quantities

• U.S. Customary System: Still used in the US. Fundamental units are foot ($ft$), slug (mass), and second ($s$).

• Metric Prefixes: Used to denote powers of ten.

  • $10^{24}$ (yotta, Y), $10^{21}$ (zetta, Z), $10^{18}$ (exa, E), $10^{15}$ (peta, P), $10^{12}$ (tera, T), $10^{9}$ (giga, G), $10^{6}$ (mega, M), $10^{3}$ (kilo, k), $10^{-1}$ (deci, d), $10^{-2}$ (centi, c), $10^{-3}$ (milli, m), $10^{-6}$ (micro, $\mu$), $10^{-9}$ (nano, n), $10^{-12}$ (pico, p), $10^{-15}$ (femto, f), $10^{-18}$ (atto, a), $10^{-21}$ (zepto, z), $10^{-24}$ (yocto, y).

• Derived Quantities: Combinations of fundamental quantities.

  • Area: Product of two lengths ($L^2$).

  • Speed: Ratio of length to time ($L/T$).

  • Density ($\rho$): Mass per unit volume. $\rho = \frac{m}{V}$.

  • Aluminum density: $2.70 \times 10^{3}\,kg/m^3$

  • Iron density: $7.86 \times 10^{3}\,kg/m^3$

Matter and Model Building

• Purpose of Models: When physicists cannot interact with a system directly (e.g., atoms), they create a mental/mathematical system of components and predict behavior based on interactions.

• History of Atomic Model:

  • Atomos: Greek term for "not sliceable," proposed by Leucippus and Democritus.

  • Discovery of Electron (1897): J.J. Thomson identified internal structure within atoms.

  • Discovery of Nucleus (1911): Nucleus identified at the center of the atom.

  • Protons and Neutrons (Early 1930s): Nucleus composed of protons (positive) and neutrons (neutral).

• Quarks: Protons and neutrons are composed of quarks. There are six varieties: Up ($u$), Down ($d$), Strange ($s$), Charmed ($c$), Bottom ($b$), and Top ($t$).

  • Charge of Up, Charmed, Top quarks: $+\frac{2}{3}$ that of a proton.

  • Charge of Down, Strange, Bottom quarks: $-\frac{1}{3}$ that of a proton.

  • Proton Composition: Two Up quarks and one Down quark ($uud$).

  • Neutron Composition: Two Down quarks and one Up quark ($ddu$).

Dimensional Analysis

• Dimension: Denotes the physical nature of a quantity (Length $L$, Mass $M$, Time $T$).

• Notation: Brackets are used to show dimensions: $[v] = L/T$, $[A] = L^2$, $[a] = L/T^2$.

• Rules of Dimensional Analysis:

  • Quantities can be added or subtracted only if they have the same dimensions.

  • Both sides of an equation must have the same dimensions.

• Applying Power Laws: To determine unknown exponents in an expression like $x = ka^n t^m$, equate the dimensions of each side:

  • $L = (L/T^2)^n T^m$

  • $L^1 = L^n T^{m-2n}\n * Equating exponents: n=1$;$m-2n=0 \rightarrow m=2$. Thus,$x = at^2.\n* **Note**: Dimensional analysis cannot determine numerical constants of proportionality (e.g., that x = \frac{1}{2}at^2).\n\n# Conversion of Units\n\n* **Algebraic Treatment**: Units are treated as algebraic quantities that cancel.\n* **Conversion Factors**:\n * 1\,mile = 1\,609\,m = 1.609\,km\n * 1\,m = 39.37\,in. = 3.281\,ft\n * 1\,ft = 0.3048\,m = 30.48\,cm\n * 1\,in. = 0.0254\,m = 2.54\,cm (exactly)\n* **Pitfall Prevention**: Always include units in every step of a calculation to detect errors where expected units are not achieved.\n\n# Estimates and Order-of-Magnitude Calculations\n\n* **Definition**: Values expressed in scientific notation as a power of ten to represent an approximate scale.\n* **Methodology**:\n 1. Express number in scientific notation (multiplier \times 10^n).\n 2. If multiplier is < 3.162$($\sqrt{10}$), the order of magnitude is$10^n.\n 3. If multiplier is > 3.162$, the order of magnitude is$10^{n+1}.\n* **Example: Breaths in a Lifetime**:\n * Assumptions: 70\,year$life,$10\,breaths/min.\n * Minutes in a year: 1\,yr \times 400\,days/yr \times 25\,h/day \times 60\,min/h \approx 6 \times 10^5\,min/yr.\n * Minutes in lifetime: 70\,yr \times 6 \times 10^5\,min/yr \approx 4 \times 10^7\,min.\n * Breaths in lifetime: 10\,breaths/min \times 4 \times 10^7\,min \approx 4 \times 10^8\,breaths.\n * Order of magnitude: 10^9\,breaths.\n\n# Significant Figures\n\n* **Uncertainty**: The number of significant figures reflects the precision of a measurement, including the first estimated digit.\n* **Zeros Hierarchy**:\n * Zeros to position decimal points (e.g., 0.03) are not significant.\n * Zeros after other digits (e.g., 1\,500$) are ambiguous unless expressed in scientific notation ($1.50 \times 10^3 has three sig figs).\n* **Mathematical Operations**:\n * **Multiplication/Division**: The result has the same number of significant figures as the quantity with the fewest significant figures in the calculation.\n * **Addition/Subtraction**: The result has the same number of decimal places as the quantity with the fewest decimal places.\n* **Rounding Rules**:\n * If the last digit dropped is > 5, round up.\n * If the last digit dropped is < 5, keep the last digit as is.\n * If the last digit dropped is exactly 5$$, round the remaining digit to the nearest even number to avoid error accumulation.

• Symbolic Solutions: It is best to solve problems algebraically first and substitute numerical data at the final step to minimize rounding errors.