Unit 1: Physical Quantities and Measurement Study Guide

Introduction to Physics and Measurement

Definition of Physics: Physics is the most fundamental of all natural sciences. It involves the study of matter, energy, and their interaction. The laws and principles of physics are essential for understanding nature.
Interdisciplinary Connections: * Biology: Uses physical principles of fluid movement to understand blood flow through the heart, arteries, and veins. * Chemistry: Relies on the physics of subatomic particles to understand the mechanisms behind chemical reactions.
Historical Context: In the nineteenth century, physical sciences were categorized into five distinct disciplines: physics, chemistry, astronomy, geology, and meteorology.
Physics and Technology: Physics forms the basis for both common and advanced technologies, including: * Everyday Devices: Computers, smartphones, MP3 players, and the internet. * Advanced Technologies: Robots (machines designed to perform tasks without human help), space shuttles, rockets, Magnetically Levitating (Maglev) trains, and microscopic robots used to fight cancer. * Medical and Military Applications: PET scans and nuclear weapons.

Branches of Physics

Physics is subdivided into various branches to accommodate its vast scope. Major branches include: * Mechanics * Optics * Oscillation and Waves * Thermodynamics * Electromagnetism * Astrophysics * Quantum Physics * Atomic Physics * Nuclear Physics * Relativity
Mathematics as a Tool: Physics has a strong connection with mathematics. Mathematical knowledge is required to understand and describe the nature of physics.
Historical Measurement Unit (Cubit): Used by Egyptians to build pyramids. A cubit is defined as the measure from the elbow to the tip of the middle finger when the arm is extended.

Physical and Non-Physical Quantities

Physical Quantities: Quantities that can be measured. Examples include length, mass, time, density, and temperature.
Non-Physical Quantities: Quantities that cannot be measured. Examples include taste, feelings, and color.
Components of Measurement: A measurement consists of two parts: 1. Numerical Magnitude: The number representing the size of the quantity. 2. Unit: The standard with which the physical quantity is compared. * Example: For a person with a height of $1.65\,\text{metres}$ ( $5\,\text{foot and 5 inches}$ ), $1.65$ is the numerical magnitude and "meter" is the unit.
Standard Comparison: Measurement is a comparison between an unknown physical quantity and a standard to determine its relative size.

Base and Derived Physical Quantities

Base (Fundamental) Quantities: These are the simplest forms of physical quantities from which all other quantities are derived. Examples include mass, length, and time.
Derived Quantities: Physical quantities obtained by multiplying or dividing base physical quantities. Examples include area ( $\text{length} \times \text{length}$ ), velocity ( $\text{length} / \text{time}$ ), and acceleration.

International System of Units (SI)

System Definition: A complete set of units for all physical quantities is called a system of units.
SI Background: "System International" (SI) is abbreviated from the French name 'System: International d' Units'.
SI Base Units: There are seven ( $07$ ) chosen base quantities, defined and standardized for accuracy and reproducibility.

SI Base Quantity	Symbol	SI Base Unit Name	Symbol
Length	$l$	meter	$m$
Mass	$m$	kilogram	$kg$
Time	$t$	second	$s$
Electric current	$I$	ampere	$A$
Temperature	$T$	Kelvin	$K$
Amount of substance	$n$	mole	$mol$
Light intensity	$I$	candela	$cd$

SI Derived Units: These are units obtained by combining base units. Some are given special names.

Derived Quantity	Symbol	SI Derived Unit Name	Symbol / Expression
Area	$A$	square meter	$m^2$
Volume	$V$	cubic meter	$m^3$
Speed, velocity	$v$	meter per second	$ms^{-1}$
Acceleration	$a$	meter per second squared	$ms^{-2}$
Density	$\rho$	kilogram per cubic meter	$kg\,m^{-3}$
Force	$F$	Newton ( $N$ )	$kg\,ms^{-2}$
Pressure	$P$	Pascal ( $Pa$ )	$kg\,m^{-1}s^{-2}$
Energy	$E, U$	Joule ( $J$ )	$kg\,m^2s^{-2}$

Standard Form and Scientific Notation

Purpose: To write very large or very small numbers compactly in powers of ten to save time and prevent errors.
General Formula: $\text{number} = \text{mantissa} \times 10^{\text{exponent}}$ * The mantissa must be a number greater than or equal to $1$ and less than $10$ .
Examples: * Width of the observable universe: $880,000,000,000,000,000,000,000,000\,m = 8.8 \times 10^{26}\,m$ . * Mass of Earth: $5,980,000,000,000,000,000,000,000\,kg = 5.98 \times 10^{24}\,kg$ . * Diameter of hydrogen nucleus: $0.0000000000000017\,m = 1.7 \times 10^{-15}\,m$ .

Prefixes to Power of Ten

Definition: A mechanism where specific names are assigned to powers of ten to simplify scientific notation.

Prefix	Symbol	Multiplier	Prefix	Symbol	Sub-multiplier
Exa	$E$	$10^{18}$	deci	$d$	$10^{-1}$
Peta	$P$	$10^{15}$	centi	$c$	$10^{-2}$
Tera	$T$	$10^{12}$	milli	$m$	$10^{-3}$
Giga	$G$	$10^{9}$	micro	$\mu$	$10^{-6}$
Mega	$M$	$10^{6}$	nano	$n$	$10^{-9}$
Kilo	$k$	$10^{3}$	pico	$p$	$10^{-12}$
Hecto	$h$	$10^{2}$	femto	$f$	$10^{-15}$
Deca	$da$	$10^{1}$	atto	$a$	$10^{-18}$

Calculations and Conversions: * Seconds in a day: $86400\,s = 8.64 \times 10^{4}\,s = 86.4 \times 10^{3}\,s = 86.4\,ks$ . * Distance to Alpha Centauri: $4.132 \times 10^{16}\,m = 41.32 \times 10^{15}\,m = 41.32\,Pm$ . * Thickness of a page: $4.0 \times 10^{-4}\,m = 0.4 \times 10^{-3}\,m = 0.4\,mm$ . * Mass of a grain of salt: $1.0 \times 10^{-3}\,g = 100 \times 10^{-3}\,g = 1.0\,mg$ (Note: Transcript error on page 11 says $10^{-3} = 100\,mg$ , likely meant $10^{-4}\,g$ for consistent conversion). * Volume Relationships: * $1\,L = 1000\,mL$ * $1\,L = 1\,dm^3 = (10\,cm)^3 = 1000\,cm^3$ * $1\,mL = 1\,cm^3$

Scalars and Vectors

Scalar Quantities: Quantities completely described by numerical magnitude and unit only. Examples: distance, speed, time, mass, energy, temperature. * Calculated using ordinary algebra (e.g., $5\,s + 20\,s = 25\,s$ ).
Vector Quantities: Quantities requiring both numerical magnitude (with unit) and direction. Examples: displacement, force, weight, velocity, acceleration, momentum, electric field strength, gravitational field strength. * Calculated using vector algebra.
Coordinate Systems: * Used to locate position using axes ( $X\text{-axis}$ and $Y\text{-axis}$ ) intersecting at an origin ( $O$ ). Position is plotted as an ordered pair $(x, y)$ .
Symbolic Representation: Represented by a letter (capital or small) with an arrow over it (e.g., $\vec{F}$ , $\vec{a}$ , $\vec{B}$ ).
Graphical Representation: Shown as an arrow. Length of the arrow represents magnitude (to scale); the arrowhead indicates direction. * Steps to represent a vector: 1. Select and draw a coordinate system. 2. Choose a suitable scale (e.g., $5\,km = 1\,cm$ ). 3. Draw a line in the fixed direction to the scaled length. 4. Add an arrowhead.
Vector Addition: The process of combining vectors into a single "resultant vector." * Requires the head-to-tail rule: Draw vectors to a common scale, place the tail of the second at the head of the first. The resultant vector joins the tail of the first to the head of the last.

Measuring Instruments

Least Count: The minimum value that can be measured on the scale of an instrument.
Metre Rule and Measuring Tape: * Metre Rule: $1\,m$ long with $1000$ small divisions (millimetres). Least count = $1\,mm$ . * Measuring Tape: Flexible ribbon (cloth, plastic, metal, fiberglass). Used for larger distances. Scales usually in inches and centimeters.
Vernier Caliper: Used to measure fractions of the smallest main scale division by sliding a second scale. * Scales: Main scale (markings of $1\,mm$ ) and Vernier scale (sliding). * Least Count Formula: $\text{Least Count} = \frac{\text{Smallest division on main scale}}{\text{Total number of divisions on vernier scale}}$ . * Example: $\frac{1\,mm}{10} = 0.1\,mm$ . * Zero Error: Occurs if zeros of main and vernier scales do not coincide when jaws are closed. * Positive: Vernier zero is to the right of main scale zero. * Negative: Vernier zero is to the left of main scale zero. * Digital Vernier Caliper: Higher precision; least count is $0.01\,mm$ .
Screw Gauge (Micrometer): Measures even smaller lengths by rotating a circular scale over a linear scale. * Pitch: Distance traveled by circular scale on linear scale in one rotation. * Least Count Formula: $\text{Least Count} = \frac{\text{Pitch of Screw Gauge}}{\text{Total number of divisions on circular scale}}$ . * Example: $\frac{0.5\,mm}{50} = 0.01\,mm$ . * Ratchet: Used to prevent excessive pressure on the object during measurement.
Physical Balance: Sensitive instrument for measuring mass to the milligram order. Consists of a vertical pillar, horizontal beam, knife edges, and two pans.
Measuring Cylinder: Used for liquid volumes or irregular solids (displacement method). Usually made of glass or plastic with scales in $mL$ or $cm^3$ . Typical least count = $1\,cm^3$ .
Stop Watch: Measures time intervals. * Mechanical (Analogue): Started/stopped via a top knob. Least count = $1\,s$ . * Digital: Usually controlled by two buttons. Least count = $0.1\,s$ .

Errors, Precision, and Accuracy

Error: The uncertainty that arises during measurement. All measurements are approximate.
Systematic Errors: Consistent errors in one direction (positive or negative). * Sources: Instrumental (imperfect design/calibration), experimental technique (environmental changes like humidity/wind), or personal bias (carelessness). * Reduction: Better instruments, improved techniques, personal care.
Random Errors: Unpredictable and irregular fluctuations. * Sources: Reaction time, technique variability, environmental fluctuations. * Reduction: Repeating measurements multiple times and calculating the mean. * Example (Pendulum): To find the period ( $T$ ), one measures the time for $10$ oscillations and divides by $10$ to reduce human reflex error.
Precision: Consistency and repeatability of measurements (degree of agreement between repeated results).
Accuracy: How close a measured value is to the true or accepted value (absence of systematic bias).
Dartboard Comparison: * Accurate and Precise: Darts hit bullseye and are grouped tightly. * Accurate not Precise: Darts near center but scattered. * Precise not Accurate: Darts grouped tightly but off-center.

Significant Figures

Definition: All accurately known digits plus the first doubtful (estimated) figure.
General Rules: 1. All reported measurement digits are significant. 2. Nonzero digits ( $1-9$ ) are always significant. 3. In numbers > 1, trailing zeros following a nonzero digit (used as placeholders) are NOT significant unless specified by scientific notation (e.g., in " $29000$ ", only $2$ and $9$ are significant). 4. In numbers < 1, zeros after the decimal but before the first nonzero digit are NOT significant (placeholders). 5. Zeros after a nonzero digit in a decimal number ARE significant (e.g., $0.00290$ has $3$ significant figures).
Rounding Rules: * If the next digit is < 5, round down (leave as is). * If the next digit is $\ge 5$ , round up (add $+1$ to the digit).

Questions & Discussion

Question: What is the least count of a metre rule that is marked with inches and feet? * Context: While the metric least count is $1\,mm$ , the Imperial least count would depend on the smallest division of the inch scale (often $\frac{1}{16}$ or $\frac{1}{32}$ of an inch).
Question: How many significant digits are in $57,000$ books? Will it change if measured in packets of $10$ ? * Context: If $57,000$ is an exact count, all digits are significant. If estimated, trailing zeros are placeholders. Measuring in packets of $10$ might change the precision/certainty of the trailing zero.
Example 1.1 Solution: * Distance between Earth and Sun ( $149,530,000,000\,m$ ): Move decimal $11$ places left $= 1.4953 \times 10^{11}\,m$ . * Mass of hair ( $0.0008\,g$ ): Move decimal $4$ places right $= 8 \times 10^{-4}\,g$ . * Seconds in a day: $24 \times 60 \times 60 = 86,400\,s = 8.64 \times 10^4\,s$ .
Example 1.2 Solution: * One ton in grams: $1000\,kg \rightarrow 1,000,000\,g = 1.0 \times 10^6\,g = 1.0\,Mg$ . * Proton diameter: $0.0000000000000017\,m = 1.7 \times 10^{-15}\,m = 1.7\,fm$ .