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1.1 Physical Quantities & Measurement — Deep, Detailed Version
Why measurement matters in physics
Physics studies how nature behaves. To move from verbal descriptions to quantitative predictions we must assign numbers to physical properties. Measurement is the bridge: it compares an unknown quantity to a known standard (a unit), producing values that can be calculated, compared and tested experimentally.
Example: Saying ‘the rod is long’ is different from ‘the rod is 1.23 m long’ — the latter supports calculations, comparisons, and error analysis.
Definitions — crisp and useful
Physical quantity: any property of a system that can be measured and expressed as a number with units (e.g., length, mass, time, temperature).
Measurement: the process of comparing a physical quantity to a standard unit (e.g., using a ruler or balance).
Unit: a standardized quantity used as a reference (meter, second, kilogram, …).
Basic (base) and derived physical quantities
In the International System (SI) all physical quantities can be expressed in terms of seven base quantities and their units. Other quantities are derived from these by algebraic combinations.
SI base quantities and their units
| Quantity | SI unit | Symbol | Brief definition (modern) |
|---|---|---|---|
| Length | metre | \(\mathrm{m}\) | Distance light travels in vacuum in \(1/299\,792\,458\) of a second. |
| Mass | kilogram | \(\mathrm{kg}\) | Defined by the Planck constant \(h\) (fixed numerical value) using the SI redefinition (quantum definition). |
| Time | second | \(\mathrm{s}\) | Duration of \(9\,192\,631\,770\) periods of radiation of Cs-133 transition. |
| Electric current | ampere | \(\mathrm{A}\) | Defined by elementary charge and other quantum standards (modern SI). |
| Temperature | kelvin | \(\mathrm{K}\) | Thermodynamic temperature scale defined from Boltzmann constant \(k\). |
| Amount of substance | mole | \(\mathrm{mol}\) | Amount containing \(N_A\) specified elementary entities (Avogadro number). |
| Luminous intensity | candela | \(\mathrm{cd}\) | Photometric measure defined via frequency and radiant intensity. |
Derived units (examples)
Derived quantities are built from base units. A few common examples:
- Area: \(\mathrm{m}^2\)
- Volume: \(\mathrm{m}^3\)
- Speed: \(\mathrm{m\ s^{-1}}\)
- Acceleration: \(\mathrm{m\ s^{-2}}\)
- Force: \(\mathrm{N} = \mathrm{kg\ m\ s^{-2}}\)
- Energy: \(\mathrm{J} = \mathrm{N\ m} = \mathrm{kg\ m^2\ s^{-2}}\)
Converting units — dimensional analysis (recipes & examples)
General method: multiply by conversion factors that equal 1 (they cancel units you don’t want and introduce units you do want).
Example 1 — inches → meters
Convert \(0.02\ \mathrm{in}\) to meters. Use \(1\ \mathrm{in}=0.0254\ \mathrm{m}\).
Also expressed as \(0.503\ \mathrm{mm}\) or \(508\ \mu\mathrm{m}\) (move decimal appropriately).
Example 2 — pounds → kilograms
Convert \(2500\ \mathrm{lb}\) to kilograms. Use \(1\ \mathrm{lb} = 0.45359237\ \mathrm{kg}\) (exact definition).
Tips for conversions
- Write units explicitly and cancel them algebraically — treat units like algebraic symbols.
- Be consistent with prefixes (kilo, milli, micro). \(1\ \mathrm{km}=10^3\ \mathrm{m}\), \(1\ \mathrm{mm}=10^{-3}\ \mathrm{m}\), \(1\ \mu\mathrm{m}=10^{-6}\ \mathrm{m}\).
- Use scientific notation for very large or very small results to keep track of significant digits easily.
Uncertainty in measurements — the why and the how
No measurement is exact. Uncertainty (also historically called ‘error’) quantifies the range in which the true value is expected to lie. Understanding and propagating uncertainty is essential for responsible physics.
Types of errors
- Systematic errors: bias that shifts all measurements (e.g., miscalibrated scale). These can often be corrected by calibration.
- Random errors: scatter in repeated measurements due to instrument resolution, human reaction time, small uncontrolled variations. These can be analyzed statistically.
Estimating uncertainty from instruments
- For an analog scale (ruler) with smallest division \(d\), a common estimate for single-measurement uncertainty is \(\pm d/2\).
- For a digital instrument that displays to the smallest increment \(d\), an uncertainty of \(\pm d\) is commonly assumed (unless otherwise specified).
Notation
Report measurements as \(x_{\text{best}} \pm \sigma\), where \(\sigma\) is the uncertainty (same units as \(x\)). Example: \(L = 5.7\ \mathrm{cm} \) (meter stick) implies \(L = 5.70 \pm 0.05\ \mathrm{cm}\) if you assume half the smallest division (0.1 cm) as uncertainty.
Worked demonstration: meter stick reading
Suppose you read a length as \(5.7\ \mathrm{cm}\) on a meter stick whose smallest division is \(0.1\ \mathrm{cm}\). Uncertainty estimate:
Significant figures — how many digits to report?
Significant figures (sig figs) communicate the precision of a measurement. Use them consistently to avoid implying false precision.
Rules (practical)
- All non-zero digits are significant (e.g. 123 → 3 sig figs).
- Zeros between non-zero digits are significant (e.g. 1002 → 4 sig figs).
- Leading zeros are not significant (e.g. 0.0025 → 2 sig figs).
- Trailing zeros in a decimal number are significant (e.g. 4.00 → 3 sig figs).
- For numbers like 300, use scientific notation to show sig figs: \(3.00\times10^2\) (3 sig figs) or \(3\times10^2\) (1 sig fig).
Arithmetic with sig figs (rules of thumb)
- Multiplication/division: result has the same number of sig figs as the factor with the fewest sig figs.
- Addition/subtraction: result has the same number of decimal places as the term with the fewest decimal places.
Multiplication example: \( (4.56)(1.4) = 6.384 \). The factor with fewer sig figs is 1.4 (2 sig figs), so report result as \(6.4\) (2 sig figs).
Addition example: \(9.65 + 8.4 – 2.89 = 15.16\). The least precise term is 8.4 (one decimal place), so round result to one decimal place → \(15.2\).
Vector vs. scalar — essential distinction
Some physical quantities have direction (vectors) while others do not (scalars). This distinction matters because vector quantities follow vector algebra (addition via components or graphical methods) while scalars obey ordinary algebra.
- Scalars: mass, time, temperature, energy, speed (magnitude only).
- Vectors: displacement, velocity, acceleration, force, momentum (magnitude + direction).
Short note on units and vectors
Units attach to magnitudes only — vector quantity example: \(\vec{v} = 3.0\ \mathrm{m\ s^{-1}}\ \hat{\mathbf{i}}\). The \(\mathrm{m\ s^{-1}}\) unit applies to the magnitude component(s).
Activity (practice problems)
- Conversion practice: A city speed limit is \(30\ \mathrm{km/h}\). Convert to miles per hour and to meters per second. (Use \(1\ \mathrm{mi}=1.60934\ \mathrm{km}\), \(1\ \mathrm{km}=1000\ \mathrm{m}\), \(1\ \mathrm{h}=3600\ \mathrm{s}\)).
- Volume unit conversion: How many cubic meters are in \(250{,}000\ \mathrm{cm^3}\)? (Recall \(1\ \mathrm{m}=100\ \mathrm{cm}\)).
- Temperature conversion: Cat average temperature \(101.5^\circ\mathrm{F}\). Convert to \(^{\circ}\mathrm{C}\). (Use \(T_C = (T_F-32)\times \tfrac{5}{9}\)).
Answers (brief):
1) \(30\ \mathrm{km/h} = 30/1.60934 \approx 18.64\ \mathrm{mi/h}\). In m/s: \(30\ \mathrm{km/h} = 30{,}000/3600 \approx 8.333\ \mathrm{m/s}\).
2) \(250{,}000\ \mathrm{cm^3} = 250{,}000\times 10^{-6}\ \mathrm{m^3} = 0.25\ \mathrm{m^3}\). (Because \(1\ \mathrm{cm^3}=10^{-6}\ \mathrm{m^3}\)).
3) \(101.5^\circ\mathrm{F} \rightarrow (101.5-32)\times 5/9 = 38.611\ldots^\circ\mathrm{C} \approx 38.6^\circ\mathrm{C}.\)
Practical tips & common student pitfalls
- Always track units: carry units through algebra — they help spot algebraic mistakes.
- Convert masses to SI: kilograms for momentum/force/energy problems.
- Do not over-report precision: round your final answer to reflect the true uncertainty.
- Be careful with prefixes: milli vs micro can differ by \(10^3\) — a costly slip.
- When in doubt, use scientific notation: it reduces human error and clarifies sig figs.
1.2 Uncertainty in Measurement & Significant Digits
Converted & expanded from the uploaded PDF (Chapter 1). :contentReference[oaicite:1]{index=1}
Why Uncertainty Matters
Every measured quantity carries uncertainty. Instruments are imperfect, readings vary, and the observer cannot mark infinitely precise digits. Physics acknowledges this by stating every measurement as:
This uncertainty tells how reliable a value is — it is the foundation of scientific honesty.
1. Sources of Error
1. Systematic Errors
These shift all readings the same way and cannot be reduced by averaging.
- Poor instrument calibration
- Zero error (e.g., scale not reading zero)
- Consistent parallax error
2. Random Errors
These fluctuate around the true value due to uncontrollable variations.
- Human reaction time
- Instrument vibration
- Limited resolution (smallest division)
2. Uncertainty From Measuring Instruments
When reading an instrument, the uncertainty comes from its resolution — the smallest division it can measure.
Rules of Thumb
- Analog scale (ruler, meter stick): Uncertainty = ±(smallest division / 2)
- Digital display: Uncertainty = ±(last displayed digit)
Example:
You read a length as \(5.7\ \mathrm{cm}\) on a meter stick with smallest division = 0.1 cm.
So the measurement is:
3. Absolute, Relative, and Percent Uncertainty
Absolute Uncertainty
The ± uncertainty in the same units as the measurement.
Relative Uncertainty
Percent Uncertainty
Example: Mass = \(50.0 \pm 0.1\ \mathrm{g}\)
Percent = \(0.002\times100 = 0.2\%\)
4. Propagation of Uncertainty
When measurements are combined by arithmetic operations, uncertainties combine too.
Addition & Subtraction
Add absolute uncertainties:
Multiplication & Division
Add relative uncertainties:
Powers
Example: Area of rectangle \(A = L \times W\)
- \( L = 10.0 \pm 0.1\ \mathrm{cm} \)
- \( W = 5.0 \pm 0.1\ \mathrm{cm} \)
Relative uncertainty = \( \frac{0.1}{10.0} + \frac{0.1}{5.0} = 0.01 + 0.02 = 0.03 \)
\( \Delta A = 0.03 \times 50 = 1.5\ \mathrm{cm^2} \)
Final result: \(A = 50.0 \pm 1.5\ \mathrm{cm^2}\)
5. Significant Digits (Significant Figures)
Significant digits express how precise a measurement is. They prevent reporting more digits than justified.
Rules for Counting Significant Figures
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros are not significant.
- Trailing zeros in decimals are significant.
- Trailing zeros in integers are ambiguous — use scientific notation.
Operations with Significant Figures
Multiplication / Division
Result has the same number of significant figures as the factor with the fewest.
Addition / Subtraction
Result has the same number of decimal places as the least precise term.
6. Reporting Measurements Correctly
Always report a measured quantity as:
Examples:
- \( 5.70 \pm 0.05\ \mathrm{cm} \)
- \( 9.81 \pm 0.03\ \mathrm{m/s^2} \)
7. Practice Problems
- Read a ruler value of \(12.4\ \mathrm{cm}\); smallest division = \(0.1\ \mathrm{cm}\). Write the measurement with uncertainty.
- A mass is measured as \(150.0\pm0.5\ \mathrm{g}\). Compute percent uncertainty.
- Given \(L = 20.0\pm0.1\ \mathrm{cm}\) and \(W = 10.0\pm0.1\ \mathrm{cm}\), compute area with uncertainty.
Solutions (brief)
- \( 12.40 \pm 0.05\ \mathrm{cm} \)
- \( \frac{0.5}{150.0}\times100 = 0.33\% \)
- \( A = 200\ \mathrm{cm^2} \); relative uncertainty = 0.005 + 0.01 = 0.015 → \( \Delta A = 3\ \mathrm{cm^2} \)
1.3 Vectors — Detailed & Expanded
Converted & expanded from your uploaded PDF (Chapter 1). :contentReference[oaicite:1]{index=1}
What Are Vectors?
A vector is a physical quantity that has both a magnitude and a direction. Examples: displacement, velocity, acceleration, force, momentum, electric field, magnetic field.
A scalar quantity has only magnitude (temperature, mass, time, energy).
1. Representing Vectors
Vectors are represented graphically by arrows:
- Length → magnitude
- Arrowhead → direction
- Tail → initial point
Notation
- \(\vec{A}\), \(\mathbf{A}\), or simply **A** (boldface)
- Magnitude: \( |\vec{A}| \) or \( A \)
2. Equality of Vectors
Two vectors are equal if:
- They have the same magnitude
- And the same direction
Their positions in space may differ; position does not matter — only length and direction do.
3. Vector Addition
1. Head-to-Tail Method
- Place the tail of the second vector at the head of the first.
- Draw the resultant \( \vec{R} \) from start of the first to end of the last.
2. Parallelogram Method
Place tails together and complete a parallelogram. The diagonal is the resultant.
4. Vector Subtraction
Defined as:
That is, reverse direction of \(\vec{B}\) and add to \(\vec{A}\).
5. Components of a Vector
A vector in 2D is decomposed along x and y axes:
If magnitude = \(A\) and angle from +x-axis = \(\theta\), then:
Magnitude from components
Direction
6. Adding Vectors Using Components
If:
Then the sum is:
7. Scalar (Dot) Product
Defined as:
In components:
Physical meaning
- Work: \(W = \vec{F}\cdot\vec{d}\)
- Projection of one vector onto another
8. Vector (Cross) Product
Defined as:
Direction: perpendicular to both vectors (right-hand rule).
Component form
9. Unit Vectors
A unit vector gives direction only.
In Cartesian coordinates:
- \(\hat{i}\) → +x direction
- \(\hat{j}\) → +y direction
- \(\hat{k}\) → +z direction
10. Practice Problems
- Find magnitude & direction of vector \((8,-6)\).
- Two vectors: \( \vec{A} = (4,3)\) and \( \vec{B} = (-2,5)\). Compute \( \vec{A}+\vec{B}\) and \( \vec{A}-\vec{B} \).
- Compute dot product of \((6,2,-1)\) and \((1,-2,3)\).
Solutions (brief)
- \(R=\sqrt{8^2+(-6)^2}=10\), direction = \( \tan^{-1}(-6/8)=-36.87^\circ\)
- \(\vec{A}+\vec{B}=(2,8)\), \(\vec{A}-\vec{B}=(6,-2)\)
- Dot = \(6(1)+2(-2)+(-1)(3)=6-4-3=-1\)
1.4 Unit Vector
Expanded from the uploaded PDF.
What Is a Unit Vector?
A unit vector is a vector that has a magnitude of exactly 1 and indicates only direction. It tells us *which way* a vector points without telling us *how large* the vector is.
If \(\vec{v}\) is any non-zero vector, then its corresponding unit vector \(\hat{v}\) is:
Why Are Unit Vectors Important?
In physics, directions must be specified clearly. Forces, velocities, fields, and accelerations all have direction. Using unit vectors allows us to:
- Break vectors into x-, y-, z- components
- Write 2D and 3D motion equations compactly
- Construct general vector expressions like \( \vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k} \)
- Describe directions independent of magnitude
Standard Unit Vectors in Cartesian Coordinates
In 2D and 3D coordinate systems, the following standard unit vectors define the positive directions of the coordinate axes:
- \(\hat{i}\): direction of +x-axis
- \(\hat{j}\): direction of +y-axis
- \(\hat{k}\): direction of +z-axis
These are mutually perpendicular:
Each has magnitude 1:
Expressing a Vector Using Unit Vectors
A vector in 3D space can be written as:
Here:
- \(A_x\): projection of \(\vec{A}\) on x-axis
- \(A_y\): projection on y-axis
- \(A_z\): projection on z-axis
Magnitude from Components
Unit Vector in Direction of \(\vec{A}\)
How to Construct a Unit Vector From Any Vector
Step 1 — Find the magnitude
Step 2 — Divide each component by the magnitude
Thus the unit vector is:
Unit Vector as a Direction Indicator
If a physical vector is written as:
Then:
- \(A\) gives the magnitude of the physical quantity.
- \(\hat{u}\) gives the direction.
Unit Vector Along a Line Between Two Points
If two points in space are:
- \(P_1(x_1, y_1, z_1)\)
- \(P_2(x_2, y_2, z_2)\)
Then the displacement vector from \(P_1\) to \(P_2\) is:
The corresponding unit vector is:
Practice Problems
- Find the unit vector of \( (8, 0, -6) \).
- A force of magnitude \(50\ \mathrm{N}\) acts in the direction \((3, 4, 0)\). Write the force vector in component form.
- Points A(2, 5, 1) and B(8, 1, 4). Find the unit vector from A → B.
Brief Answers
- \( |\vec{v}|=\sqrt{8^2+0+(-6)^2}=10 \Rightarrow \hat{v}=(0.8,0,-0.6) \)
- Unit direction = \( \left(\frac{3}{5},\frac{4}{5},0\right) \) ⇒ \( \vec{F} = 50\left( \frac{3}{5}\hat{i}+\frac{4}{5}\hat{j} \right) =(30, 40, 0)\,\mathrm{N} \)
- \(\vec{AB}=(6,-4,3)\), magnitude = \( \sqrt{36+16+9}=7 \) ⇒ unit vector = \( \left(\frac{6}{7},-\frac{4}{7},\frac{3}{7}\right) \)