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).