Physics – Measurement & Numerical Skills

Physics, Matter & Energy – Why Measurement Matters

• Physics studies the nature and properties of matter and energy.

  • Concerned with motion and behaviour of matter through time & space, and its relation to energy & force.

• Precise measurement underpins every physical law; wrong units ⇨ wrong conclusions.

Measurement: Core Subtopics Introduced

• 01 Conversion of Units (SI ↔ English, and vice-versa)

• 02 Rounding-Off of Numbers

• 03 Scientific Notation & Operations

• 04 Unit Prefixes

• 05 Significant Figures

  • Together these skills guarantee numerical accuracy, comparability, and reproducibility in experiments and engineering designs.

Unit Conversion – General Idea

• Definition: Changing a numerical value from one unit system to another for the same physical quantity via multiplicative factors.

• “Always multiply by a fraction that equals $1$ but carries the desired units.”

• SI base examples: meter, kilogram, kelvin, ampere, mole, candela.

• English / Imperial examples: inch, foot, yard, mile, ounce, pound.

Key SI ↔ English Conversion Factors

• Length

  • $1\;\text{inch}=2.54\;\text{cm}$ (exact by international agreement)

  • $1000\;\text{mil}=1\;\text{inch}$

  • $1\;\text{ft}=0.3048\;\text{m}$

  • $1\;\text{yd}=3\;\text{ft}$

  • $1\;\text{fathom}=6\;\text{ft}$

  • $1\;\text{chain}=66\;\text{ft}$

  • $1\;\text{furlong}=660\;\text{ft}$

  • $1\;\text{mile}=5280\;\text{ft}$

• Mass / Weight & Force

  • $1\;\text{kg}=2.21\;\text{lbm}$ (mass)

  • $1\;\text{kg}=35.274\;\text{oz}$

  • $1\;\text{slug}=32.2\;\text{lbm}$

  • $1000\;\text{kg}=1\;\text{ton}\;\text{(metric)}$

  • $1\;\text{short ton}=2000\;\text{lbm}$

  • $1\;\text{long ton}=2240\;\text{lbm}$

  • Force relations

  • $100\,000\;\text{dyn}=1\;\text{N}$

  • $1\;\text{kgf}=9.81\;\text{N}$

  • $1\;\text{lbf}=4.448\;\text{N}$

Algorithmic, Step-by-Step Conversion

1. Write the given quantity over $1$.

2. Multiply by conversion factors written as fractions so that unwanted units cancel.

3. Continue chaining factors until only the desired unit remains.

4. Multiply (or divide) numerators & denominators ⇒ final value.

Worked Examples

• Convert $2\;\text{m}$ to $\text{ft}$

  • $2\,\text{m}\times \frac{1\,\text{ft}}{0.3048\,\text{m}}=6.56\,\text{ft}$

• Convert $3\,\text{m}\rightarrow\text{cm}$

  • $3\,\text{m}\times\frac{100\,\text{cm}}{1\,\text{m}} = 300\,\text{cm}$

• Convert $7\,\text{cm}\rightarrow\text{in}$

  • $7\,\text{cm}\times\frac{1\,\text{in}}{2.54\,\text{cm}} = 2.76\,\text{in}$

• Convert $5\,\text{m}\rightarrow\text{in}$

  • $5\,\text{m}\times\frac{100\,\text{cm}}{1\,\text{m}}\times\frac{1\,\text{in}}{2.54\,\text{cm}} = 196.85\,\text{in}$

• Convert $27\,\text{lbf}\rightarrow\text{kgf}$

  1. Force to SI: $27\,\text{lbf}\times\frac{4.448\,\text{N}}{1\,\text{lbf}}$

  2. N to kgf: $\times\frac{1\,\text{kgf}}{9.81\,\text{N}} = 12.24\,\text{kgf}$

• Convert $60\;\text{mi/hr}\rightarrow\text{ft/s}$

  • $60\,\frac{\text{mi}}{\text{hr}}\times\frac{5280\,\text{ft}}{1\,\text{mi}}\times\frac{1\,\text{hr}}{3600\,\text{s}}=88\,\frac{\text{ft}}{\text{s}}$

• Convert $15,000\;\text{furlongs/fortnight}\rightarrow \text{m/s}$

  • Chain factors: furlong→ft, ft→m, fortnight→weeks→days→hours→minutes→seconds.

  • Published result: $2.49\;\text{m/s}$ (humorous yet fully valid!)

Rounding-Off Rules

• Identify the digit to keep (last significant digit).

• If digit to its right < $5$ ⇒ keep digit unchanged.

• If digit to its right ≥ $5$ ⇒ increase last kept digit by $1$.

• All digits right of the last kept digit become $0$.

Examples

Scientific Notation – Concept & Conversion

• Format: $a\times10^{n}$ where 1\le |a|<10 and $n$ is an integer.

• Converting sci-notation → standard

  • Positive $n$: move decimal $n$ places RIGHT; pad with zeros.

  • Negative $n$: move decimal $|n|$ places LEFT; pad with zeros.

  • Example: $7.2\times10^{5}=720\,000$

  • Example: $3.8\times10^{-5}=0.000038$

• Converting standard → sci-notation

  • $10,358,000 = 1.0358\times10^{7}$

  • $0.001256 = 1.256\times10^{-3}$

• Check-back exercises

  • $7.3962\times10^{3}=7\,396.2$

  • $9.2\times10^{-6}=0.0000092$

Operations in Scientific Notation

• Addition / Subtraction

  1. Rewrite numbers so they share the same exponent.

  2. Add or subtract the cited coefficients.

  • Example: $(3.48\times10^{3})+(2.36\times10^{4})=(0.348\times10^{4})+(2.36\times10^{4})=2.708\times10^{4}$

  • Example: $(5.92\times10^{5})+(8.61\times10^{4})=(5.92\times10^{5})+(0.0861\times10^{5})=6.0061\times10^{5}$

• Multiplication

  • Multiply coefficients, add exponents: $(6.72\times10^{4})(3.46\times10^{4})=23.2512\times10^{8}=2.32512\times10^{9}$

• Division

  • Divide coefficients, subtract exponents: $\frac{5.49\times10^{6}}{2.07\times10^{3}}=2.65\times10^{3}$

Unit Prefixes – Orders of Magnitude

• Prefix symbol precedes the base-unit symbol to express powers of $10$.

  • e.g.

  • $\text{pico (p)} = 10^{-12}$

  • $\text{nano (n)} = 10^{-9}$

  • \text{micro (\mu)} = 10^{-6}

  • $\text{milli (m)} = 10^{-3}$

  • $\text{kilo (k)} = 10^{3}$

  • $\text{mega (M)} = 10^{6}$

  • $\text{giga (G)} = 10^{9}$

  • $\text{tera (T)} = 10^{12}$

  • $\text{peta (P)} = 10^{15}$

  • $\text{exa (E)} = 10^{18}$

• Base units: gram, meter, hertz etc. Each by itself signifies $10^{0}$.

Conversion Formula (Prefix to Prefix)

$\text{Quantity}\times\frac{1\;\text{prefix}<em>1}{10^{m}}\times\frac{10^{n}}{1\;\text{prefix}</em>2} = \text{Quantity}\times10^{n-m}\;(\text{prefix}_2)$
where

• $m$ = exponent of original prefix

• $n$ = exponent of target prefix

Example Conversions

• $22\,\text{pF}\rightarrow\text{MF}$

  • $m=-12,\;n=6$ ⇒ shift $\Delta=6-(-12)=18$

  • $22\times10^{-18}=2.2\times10^{-17}\;\text{MF}$

• $34\,\text{Mb}\rightarrow\text{nb}$

  • $m=6,\;n=-9;\;\Delta=-9-6=-15$

  • $34\times10^{15}=3.4\times10^{16}\;\text{nb}$

• $10\,\text{fg}\rightarrow\mu\text{g}$

  • $m=-15,\;n=-6;\;\Delta= -6-(-15)=9$

  • $10\times10^{9}=1\times10^{10}\;\mu\text{g}$ (illustrative; transcript shows different rounding)

• $12\,\text{Pg}\rightarrow\text{hg}$

  • $m=15,\;n=2;\;\Delta=2-15=-13$

  • $12\times10^{-13}=1.2\times10^{-12}\;\text{hg}$

• $429\,\text{TL}\rightarrow\text{nL}$

  • $m=12,\;n=-9;\;\Delta=-9-12=-21$

  • $429\times10^{21}=4.29\times10^{23}\;\text{nL}$

Significant Figures (Sig Figs)

• Reflect the precision of a measured or calculated quantity.

• Rules for counting sig figs

  1. All non-zero digits are significant.

  2. Zeros between non-zero digits are significant.

  3. Leading zeros (left of the first non-zero digit) are NOT significant.

  4. Trailing zeros to the right of a decimal point ARE significant.

  • Trailing zeros in whole numbers with no decimal shown are ambiguous; use scientific notation for clarity.

Practice – How Many Significant Figures?

Practical, Ethical, & Real-World Relevance

• Engineering failures (Mars Climate Orbiter, NASA 1999) arose from unit mix-ups—highlighting the ethical duty of accurate conversion.

• Rounding & sig-fig awareness prevents misleading precision in lab reports.

• Scientific notation keeps data readable from nanoscopic ($10^{-9}$ m) to astronomical scales ($10^{21}$ m).

• Consistent prefixes allow global data exchange (e.g., storage: MB vs MiB confusion).