SI Units and Unit Conversion
Units and Their Importance
- Units provide a standardized way to describe how much of a physical quantity we have.
- Without units, numbers are meaningless or ambiguous (e.g.
- “5” could mean 5 \text{ m}, 5 \text{ s}, 5 \text{ kg}, etc.).
- Correct units are critical in science, engineering, commerce, and everyday life.
- Real-world consequence: The 1999 Mars Climate Orbiter failed because software from Lockheed Martin used U.S. customary pound-force (lbf) while NASA expected metric newtons (N):
1 \text{ lbf} = 4.45 \text{ N} \neq 1 \text{ N}
Mismatch led to trajectory errors and loss of the spacecraft.
What Is a Unit?
- A unit is “a way to define a specific amount of something.”
Example: meter defines a specific amount of length; second defines a specific amount of time. - We can combine simple (base) units to form derived units (e.g., \dfrac{\text{distance}}{\text{time}} = \text{speed}).
Everyday (Non-SI) Units
| Physical Quantity | Common Units | Abbreviations |
|---|---|---|
| Length | centimeter, inch, foot, mile | cm, in, ft, mi |
| Time | second, minute, hour | s, min, hr |
| Temperature | Fahrenheit, Celsius | °F, °C |
| Volume | ounce, cup, liter | oz, cup, L |
| Money | dollar, cent | $ , ¢ |
Combining Units in Daily Life
| Derived Quantity | Example Units | Interpretation |
|---|---|---|
| Speed | miles/hour (mph), km/h | \dfrac{\text{distance}}{\text{time}} |
| Fuel efficiency | miles/gallon (mpg), km/L | \dfrac{\text{distance}}{\text{volume}} |
| Heart rate | beats/minute (bpm) | \dfrac{\text{beats}}{\text{time}} |
Units in Stated Physics Problems
- “A rock is dropped off a cliff that is 65 feet (ft) high…” vs. “…65 feet high.”
Always retain units explicitly. - “A car is driving 18 miles/hour…” vs. “…18 mph.”
Units remind us how quantities relate (distance per time).
ALWAYS Keep Track of Your Units
- Write units beside every number through all algebraic steps.
- Perform algebra on symbols and units together to catch mismatches early.
The International System of Units (SI)
- “Système International” – globally agreed standard; Physics typically defaults to SI.
The 7 Base SI Units
| Physical Quantity | SI Unit | Symbol |
|---|---|---|
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Electrical current | ampere | A |
| Luminous intensity | candela | cd |
Forming Derived SI Units (Selected Examples)
| Derived Quantity | Algebraic Combination | Derived SI Name / Symbol |
|---|---|---|
| Speed / Velocity | \dfrac{\text{m}}{\text{s}} | m·s⁻¹ |
| Volume | \text{m}^3 | cubic meter |
| Density | \dfrac{\text{kg}}{\text{m}^3} | kg·m⁻³ |
| Force | \dfrac{\text{kg·m}}{\text{s}^2} | newton (N) |
(The slides repeated the derived-unit table on pages 14–17 for emphasis.)
Converting Between Units (Dimensional Analysis)
Fundamental Principle
- Identify equal-amount relationships (conversion factors). Example:
60 \text{ s} = 1 \text{ min}, 1 \text{ km} = 0.62 \text{ mi} - Multiply by fractions built from these equalities so that unwanted units cancel and desired units remain.
- Track every factor explicitly (often arranged as “Starting Amount → × Equal Amounts → Final Amount”).
Worked Examples
1. Length of a Textbook
- Given: 9 \text{ in} = 22.9 \text{ cm} (measured). Illustrates direct equivalence.
2. Adding Mixed Units (4 in + 15 cm)
- Convert 15 cm → inches using 1 \text{ in} = 2.54 \text{ cm}:
15 \text{ cm} \times \dfrac{1 \text{ in}}{2.54 \text{ cm}} = 5.9 \text{ in} - Add: 4 \text{ in} + 5.9 \text{ in} = 9.9 \text{ in}.
- OR convert 4 in → cm and add:
4 \text{ in} \times \dfrac{2.54 \text{ cm}}{1 \text{ in}} = 10.2 \text{ cm}
10.2 \text{ cm} + 15 \text{ cm} = 25.2 \text{ cm}. - Result: 9.9 \text{ in} \equiv 25.2 \text{ cm}.
3. Seconds in a Day
Relationships:
60 \text{ s} = 1 \text{ min},\; 60 \text{ min} = 1 \text{ hr},\; 24 \text{ hr} = 1 \text{ day}
Dimensional-analysis string:
1 \text{ day}\times\dfrac{24 \text{ hr}}{1 \text{ day}}\times\dfrac{60 \text{ min}}{1 \text{ hr}}\times\dfrac{60 \text{ s}}{1 \text{ min}} = 86{,}400 \text{ s}
4. Convert 25 mph to km/h
Relationship: 1 \text{ km} = 0.62 \text{ mi}
25 \dfrac{\text{mi}}{\text{hr}}\times\dfrac{1 \text{ km}}{0.62 \text{ mi}}\approx 40.3 \dfrac{\text{km}}{\text{hr}}
Result: 25 \text{ mph} \approx 40.3 \text{ km/h}.
Dimensional Analysis Template (Slide Diagram)
Starting Amount → multiply by one or more Equal Amounts (conversion factors) until desired units are obtained → Final Amount.
Graphically (as shown on slides 31–33 & 44):
\text{Starting}\;A\times\frac{B}{A}\times\frac{C}{B}\times\cdots = \text{Final}\;C
Each numerator equals its denominator, so the numerical value changes but the physical quantity stays the same.
Sample Review / Practice Problems (Slides 42–43)
- Gravitational attraction between 400 kg and 750 kg objects 80 m apart—find F.
- Two-planet problem with radii, distance, and known force multiplier—solve for mass of Planet A.
- Orbit time of the Moon given Earth radius—all reinforce correct-unit use (answer keys not provided in transcript).
Key Takeaways
- Units communicate what a number measures; they must accompany every numerical result.
- SI provides seven base units from which all other scientific units derive.
- Unit conversion uses dimensional analysis: multiply by fractions equal to 1 so unwanted units cancel.
- Always verify units algebraically—failure to do so can destroy multimillion-dollar missions (Mars Climate Orbiter).