General Physics 1 – Physical Quantities & Measurements

Lesson 2.1 – Accuracy and Precision

• Physical measurement = quantitative comparison with a reference; never perfectly exact.

• Two complementary qualities:
• Accuracy – degree to which a measured value agrees with the true (accepted) value.
• Precision (repeatability / reproducibility) – degree to which repeated measurements agree with one another.

• Uncertainty / Limit of Probability
• Expresses the spread within which the true value is expected to lie.
• Written “± value” after the measurement.
• Percent-form formula:
$\text{Percent Uncertainty} = \frac{\text{Limit of Probability}}{\text{Given Value}} \times 100\%$

• Worked example 1 – wooden beam
• Reported: $220\,\text{cm} \pm 5\,\text{cm}$.
• Percent uncertainty: $\frac{5}{220}\times100\% = 2.27\%$.
• Final statement: $220\,\text{cm}\,\pm 2.27\%$.

• Worked example 2 – bag of flour on digital scale

1. Range: $500-10=490\,\text{g} \;\text{to}\; 500+10=510\,\text{g}$.
2. Percent uncertainty: $\frac{10}{500}\times100\% = 2\%$.
3. Statement: $500\,\text{g}\,\pm 2\%$.

• Practice items (compute percent uncertainty & give final form)

1. $300\,\text{cm} \pm 5\,\text{cm}$
2. $460\,\text{cm} \pm 20\,\text{cm}$
3. $360\,\text{cm} \pm 35\,\text{cm}$

• Visual summary (bull’s-eye diagrams)
• Low accuracy / Low precision – points spread & far from true value.
• Low accuracy / High precision – clustered points offset from true value.
• High accuracy / Low precision – points scattered around bull’s-eye.
• High accuracy / High precision – tight cluster centred on bull’s-eye.

• Key take-away: More significant digits ⇒ potentially higher accuracy but only if measurement process is also accurate; repeated trials reveal precision.

Lesson 2.2 – Scientific Notation

• Shorthand (power-of-ten) representation for very large or very small numbers.

• Correct format conditions

1. Coefficient $a$: 1 \le a < 10.
2. Base $=10$.
3. Exponent $b$: integer showing decimal-place shift.
   • b>0 ⇒ original number ≥ 10; decimal moved left.
   • b<0 ⇒ original number < 1; decimal moved right.

• General form: $N = a \times 10^{b}$

• Conversion procedure

1. Move decimal so only one non-zero digit remains left of decimal.
2. Count places moved ⇒ exponent magnitude.
3. Direction of move sets sign of exponent.

• Examples
• $3\,250\,000\,000 = 3.25 \times 10^{9}$ (moved 9 left → $b=+9$).
• $0.0000004 = 4 \times 10^{-7}$ (moved 7 right → $b=-7$).
• Quick reference powers: $10^{0}=1,\;10^{3}=1{,}000,\;10^{-3}=0.001$ etc.

• Back-conversion: shift decimal right for positive $b$, left for negative $b$.
• $6.03 \times 10^{7}=60{,}300{,}000$.
• $5.3 \times 10^{-4}=0.00053$.

• Operations
• Multiplication: multiply coefficients, add exponents.
$\bigl(2.0 \times 10^{4}\bigr)(4.0 \times 10^{2}) = 8.0 \times 10^{6}$.

• Division: divide coefficients, subtract exponents.
$\frac{4.0 \times 10^{2}}{2.0 \times 10^{4}} = 2.0 \times 10^{-2}$.

• Addition / subtraction (review of signed numbers): first express numbers with the same exponent, then operate on coefficients.

• Sample multiplication task structure (given for students):

1. Multiply decimal parts.
2. Handle powers of ten.
3. Re-express in scientific notation if necessary.

• Exercises to practice
• Write in scientific notation: $0.000005,\;3{,}000{,}000{,}000,\;0.0056$.
• Write in standard form: $5.2 \times 10^{3},\;9.65 \times 10^{-4},\;8.5 \times 10^{-2}$.
• Mixed conversions list (78 000, 0.00053, …)

Lesson 2.3 – Significant Figures

• Significance: Measurement data are meaningful only when reported with the correct significant digits (sig figs); more sig figs typically ⇒ more accuracy.

• Determining sig figs – rules on zeros

1. All non-zero digits are significant.
2. Zeros between significant digits are significant.
   • Example: $30.7$ has 3 sig figs.
3. Leading zeros (before first non-zero digit) are not significant—only locate decimal.
   • Example: $0.0678$ has 3 sig figs.
4. Trailing zeros after a decimal point are significant.
   • Example: $20.00$ has 4 sig figs.
5. Trailing zeros in a whole number without a decimal are not presumed significant unless a bar, underline or decimal is shown.
   • Example: $1000$ has 1 sig fig; $1000.$ has 4 sig figs.

• Illustrative set
• $345.72$ → 5 sig figs.
• $0.0028$ → 2.
• $10.05$ → 4.
• $1000$ → 1; $1000.$ → 4.
• $2.50$ → 3.
• $0.0709$ → 3.
• $50600$ → 3.

• Seatwork examples (answers given in slides)

1. $3100.0 \times 10^{2}$ → 5 sig figs.
2. $0.982 \times 10^{-3}$ → 3.
3. $20.4 \times 10^{4}$ → 3.
4. $(-4.12 \times 10^{-4})(7.33 \times 10^{12}) = -3.04 \times 10^{9}$ (answer maintains 3 sig figs).
5. $\frac{1.0 \times 10^{-14}}{4.6 \times 10^{-6}} = 2.18 \times 10^{-9}$ (3 sig figs).

Lesson 2.3 – System of Units (SI & Conversions)

• SI (Système International d’Unités)—metric-based, seven fundamental units.
• Three primary in mechanics: mass (kg), length (m), time (s).

• Derived physical quantities: area, volume, density, speed, acceleration, power, energy, pressure, viscosity, etc.

• Prefixes for Powers of Ten (selected list)
• $10^{12}$ tera (T)
• $10^{9}$ giga (G)
• $10^{6}$ mega (M)
• $10^{3}$ kilo (k)
• $10^{-2}$ centi (c)
• $10^{-3}$ milli (m)
• $10^{-6}$ micro ($\mu$)
• $10^{-9}$ nano (n)
• $10^{-12}$ pico (p)

• Common length equivalents
• $1\,\text{km}=1000\,\text{m}$
• $1\,\text{dm}=0.1\,\text{m}=10\,\text{dm per m}$
• $1\,\text{cm}=0.01\,\text{m}=100\,\text{cm per m}$
• $1\,\mu\text{m}=10^{-6}\,\text{m}$
• $1\,\text{nm}=10^{-9}\,\text{m}$
• $1\,\text{pm}=10^{-12}\,\text{m}$

• Mass & Time fundamentals
• Mass: kilogram (kg) with same prefix system (mg, g, Mg, …).
• Time: second (s).

• Dimensional-analysis conversion strategy
$\text{Quantity}\;\times\;\frac{\text{Conversion factor}<em>1}{1}\;\times\;\frac{\text{Conversion factor}</em>2}{1}\;=\;\text{Desired unit}$

• Multi-step worked example
• Convert $1{,}250{,}000\,\text{mg}$ to kilograms.
1. $1{,}250{,}000\,\text{mg}\times\frac{1\,\text{g}}{1000\,\text{mg}}=1250\,\text{g}$
2. $1250\,\text{g}\times\frac{1\,\text{kg}}{1000\,\text{g}}=1.25\,\text{kg}$

• Convert $3\,\text{km}$ to centimetres.
$3\,\text{km}\times\frac{1000\,\text{m}}{1\,\text{km}}\times\frac{100\,\text{cm}}{1\,\text{m}}=300{,}000\,\text{cm}$

• Imperial conversion reference (partial list)
• $1\,\text{ft}=12\,\text{in}$, $1\,\text{yd}=3\,\text{ft}=36\,\text{in}$, $1\,\text{mi}=5280\,\text{ft}=1760\,\text{yd}=63{,}360\,\text{in}$.
• $1\,\text{lb}=16\,\text{oz}=0.000453\,\text{T (ton)}$.
• Fluid: $1\,\text{gal}=4\,\text{qt}=8\,\text{pt}=16\,\text{c}$.

• Conversion problem set (30 questions) supplied for extra practice—answers provided in slide 59 for verification (e.g., $360\,\text{in}=30\,\text{ft}$; $44\,\text{qt}=11\,\text{gal}$ …).

• Ethical / practical context
• Proper unit usage ensures universal understanding and prevents catastrophic errors (e.g., Mars Climate Orbiter lost due to unit mix-up).
• Reporting uncertainty & sig figs maintains scientific honesty.

• Connections & prerequisites
• Builds on arithmetic/algebra (percent, exponent rules).
• Foundational for later topics: kinematics (needs unit consistency), dynamics (error bars in experiments), thermodynamics, etc.

• Quick summary checklist
✔ Know difference between accuracy & precision, compute percent uncertainty.
✔ Convert to/from scientific notation, perform arithmetic with powers of ten.
✔ Apply sig-fig rules, retain correct sig figs in calculations.
✔ Memorize SI prefixes, perform dimensional-analysis conversions.
✔ Always state units and uncertainties when reporting a measurement.