Units, Velocity, and Uncertainty: Foundational Concepts in Physics I

Units and Measurement in Physics

International System of Units (SI)

The International System of Units (SI) is the globally accepted standard for measurement in science and engineering. SI units provide consistency and clarity in expressing physical quantities.

• Base Units: All other SI units are derived from these seven fundamental units:

Quantity	Unit Name	Symbol
Mass	kilogram	kg
Length	meter	m
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

• Derived Units: Created by combining base units (e.g., velocity: m/s, force: kg·m/s2).

Dimensional Analysis

Dimensional analysis is a method for converting between units and checking the consistency of equations. It is essential for solving physics problems and ensuring correct results.

• Example: Convert 60 mi/hr to m/s.

• Conversion factor: 1 mile = 1.6 kilometers

• Process: Multiply by conversion factors to cancel units.

Common Prefixes and Scales:

Prefix	Symbol	Value	Example
Micro	μ	$10^{-6}$	Size of bacteria: $1 ext{ μm} = 10^{-6} ext{ m}$
Kilo	k	$10^{3}$	1 km = $10^{3}$ m (a 10-minute walk)
Milligram	mg	$10^{-6}$ kg	Mass of a grain of salt
Gram	g	$10^{-3}$ kg	Mass of a paper clip
Nanosecond	ns	$10^{-9}$ s	Time for light to travel 0.3 m

Significant Figures and Uncertainty

Significant Figures (Sig Figs)

Significant figures reflect the precision of a measured or calculated quantity. The number of significant digits in a result should match the least precise measurement used in the calculation.

• Example: Calculating the area of a rectangle with length = 12 mm (2 sig figs) and width = 5.98 mm (3 sig figs):

• Area = $12 ext{ mm} imes 5.98 ext{ mm} = 71.76 ext{ mm}^2$

• Final answer should be rounded to 2 significant figures: $72 ext{ mm}^2$

Perimeter Calculation:

• Perimeter = $2 imes (12 ext{ mm} + 5.98 ext{ mm}) = 35.96 ext{ mm}$

• Round to 2 significant figures: $36 ext{ mm}$

Length-to-Width Ratio:

• Ratio = $12 / 5.98 ightarrow 2.0$ (rounded to 2 sig figs)

Density Calculations and Significant Figures

Density is defined as mass divided by volume. The precision of the answer depends on the significant figures in the measurements.

• Formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Example: Mass = 1.80 kg, Volume = $6.0 \times 10^{-4}$ m3

• Density = $\frac{1.80}{6.0 \times 10^{-4}} = 3.0 \times 10^{3}$ kg/m3

• Depending on significant figures, answers may be written as $3 \times 10^{3}$, $3.0 \times 10^{3}$, $3.00 \times 10^{3}$, or $3.000 \times 10^{3}$ kg/m3

All these answers are mathematically equivalent, but the number of significant figures reflects the precision of the measurements.

Estimating Physical Quantities

Estimating the Weight of an Average Person

Physics often uses simplified models to estimate quantities. For example, the human body can be approximated as a rectangular prism to estimate volume and mass.

• Dimensions: Height ≈ 5.5 ft, Width ≈ 12 in, Depth ≈ 6.0 in

• Volume Calculation: Convert all dimensions to centimeters, then multiply:

• Height: $5.5 \text{ ft} \times 30.48 \text{ cm/ft} = 168 \text{ cm}$

• Width: $12 \text{ in} \times 2.54 \text{ cm/in} = 30.5 \text{ cm}$

• Depth: $6.0 \text{ in} \times 2.54 \text{ cm/in} = 15.2 \text{ cm}$

• Volume: $168 \text{ cm} \times 30.5 \text{ cm} \times 15.2 \text{ cm} \approx 78,000 \text{ cm}^3$

• Mass Estimation: Assuming the body is 100% water ($\rho = 1 \text{ g/cm}^3$):

• Mass = $78,000 \text{ cm}^3 \times 1 \text{ g/cm}^3 = 78,000 \text{ g} = 78 \text{ kg}$

• Weight Conversion: $1 \text{ kg} = 2.2 \text{ lb}$, so $78 \text{ kg} \approx 172 \text{ lb}$

Example: Estimating the mass and weight of a person using simplified geometric models is a common approach in introductory physics.

Vectors and Scalars

Definitions and Properties

Physical quantities are classified as either scalars or vectors:

• Scalars: Quantities with magnitude only (e.g., mass, speed, time, volume).

• Vectors: Quantities with both magnitude and direction (e.g., displacement, velocity, force).

Notation: Vectors are often denoted by boldface letters or an arrow above the symbol (e.g., v or $\vec{v}$).

Vector Components and Trigonometry

Vectors can be resolved into horizontal (x) and vertical (y) components using trigonometric functions.

• Given: Vector A with magnitude $A$ and angle $\theta$ with respect to the x-axis.

• Formulas:

• Horizontal component: $A_x = A \cos \theta$

• Vertical component: $A_y = A \sin \theta$

Pythagorean Theorem: Used to relate the components and magnitude of a vector:

• $A = \sqrt{A_x^2 + A_y^2}$

Trigonometric Relationships:

• $\tan \theta = \frac{A_y}{A_x}$

• $\sin \theta = \frac{A_y}{A}$

• $\cos \theta = \frac{A_x}{A}$

Example: If a vector has magnitude 5 m/s and direction 42°, its components are:

• $A_x = 5 \cos 42^\circ$

• $A_y = 5 \sin 42^\circ$

Sample Problems and Applications

Speed and Velocity Calculations

Speed is a scalar quantity representing the rate of motion, while velocity is a vector that includes direction.

• Formula for average speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Example: A person travels 1.5 miles east in 5.23 minutes, then 210 ft east in 63 seconds. Calculate speed in mi/hr and m/s.

• Convert all distances to meters and times to seconds for SI units.

• Final answers: $20 \text{ mi/hr}$, $9.0 \text{ m/s}$ (rounded appropriately)

Review Table: Scalars vs. Vectors

Quantity	Scalar	Vector
Speed	✔️
Mass	✔️
Time	✔️
Volume	✔️
Displacement		✔️
Velocity		✔️
Force		✔️

Additional info: Some context and formulas have been expanded for clarity and completeness, including step-by-step unit conversions and trigonometric relationships for vectors.