Physics: Preliminaries and Vectors

Physics Preliminaries

Chapter One: Preliminaries

• Physics originates from the Greek word meaning "nature."
• It serves as the foundation for science with numerous applications that simplify life.
• Physics studies matter in relation to energy and the precise measurement of natural phenomena.
• It's fundamentally a science of measurement.
• The principles of physics underlie engineering and technology.
• Measurement involves comparing an unknown quantity to a known standard unit.
• A unit is a standard quantity used for measurement (e.g., kilogram).

Learning Objectives:

• Explain physics.
• Describe SI base units.
• Describe derived units.
• Use metric prefixes with SI units.
• Describe models, theories, and laws.
• Know units for physical quantities.
• Use prefixes for quantity scaling.
• Apply unit conversion (dimensional analysis).
• Understand uncertainty and significant figures.

1.1 Physical Quantities and Measurement

• Self Diagnostic Test:
  • Why measurement is needed in physics and daily life.
  • Names and abbreviations of basic physical quantities and SI units.
  • Definition of a unit.

Definitions:

• Physical quantity: A quantifiable property of a phenomenon or body (e.g., length, mass).
• Measurement: Comparing a physical quantity to a standard.
• New physical quantities can be created as needed.
• Basic Physical Quantities: Fundamental quantities from which others are derived (e.g., length, mass, time).
• Derived Physical Quantities: Quantities expressed in terms of basic quantities (e.g., area, volume, density).
• Numerical values and equations enhance understanding of nature.
• All physical quantities are combinations of seven basic physical quantities.
• Physical quantities are defined by measurement methods or calculation from other measurements.

1.1.1 Physical Quantities

A. Basic Physical Quantities:

• Cannot be expressed in terms of other physical quantities.
• Examples: length, mass, and time.

B. Derived Physical Quantities:

• Expressed in terms of fundamental quantities.
• Examples: area, volume, and density.
• Units are standardized values for expressing measurements.

1.1.2 SI Units: Basic and Derived Units

• SI (International System of Units): Modern metric system agreed upon in 1960.
• Globally adopted as the primary system of units.
• Built on 7 basic quantities and associated units.

1.1.3 Conversion of Units

• Multiply by conversion factors to change units.

Examples:

1. Length: $0.02 \text{ in} = 0.02 \times 0.0254 \text{ m} = 0.000508 \text{ m} = 5.08 \times 10^{-4} \text{ m} = 0.508 \text{ mm} = 508 \mu \text{m}$
2. Weight: $2500 \text{ lb} = 2500 \times 0.4536 \text{ kg} = 1134.0 \text{ kg}$

1.2 Uncertainty in Measurement and Significant Digits

• Measurements have inherent uncertainty.
• Uncertainty analysis estimates deviations from the true value.
• Uncertainty indicates the range of possible values with a confidence level.
• All measurements have errors, categorized as:
  1. Systematic Error: Calibration errors, consistently too small or large, eliminated by pre-calibration.
  2. Random Errors: Fluctuations around the average, equally probable to be too large or small, due to scale limitations.
• Statistical methods are needed for random uncertainty analysis.

Rules of thumb for uncertainty in single measurements:

1. Scale measuring device: Uncertainty = smallest increment $\div 2$. Example: Meter Stick.
2. Digital measuring device: Uncertainty = smallest increment. Example: Digital Balance reading 5.7513 kg.

• Stating Measurements:
  • Explicitly state uncertainty.
  • Implied uncertainty: e.g., 5.7 cm implies $5.65 \text{ cm} \leq L \leq 5.75 \text{ cm}$, uncertainty of 0.05 cm.
  • General form: Measurement = $x_{\text{best}} \pm \text{uncertainty}$
  • $x_{\text{best}}$ = best estimate

1.2.1 Significant Digits

• Meaningful digits imply measurement error.
• Example: 0.428 m implies $\pm 0.001 \text{ m}$
• Report significant figures consistent with estimated error.
• 0.428 m has three significant digits.
• Convention: Report only one uncertain digit.
• Example: If estimated error is 0.02 m, report $0.43 \pm 0.02 \text{ m}$
• Zeros:
  • If a zero has a non-zero digit to its left, it is significant.
  • 5.00 has 3 significant figures.
  • 0.0005 has 1 significant figure.
  • 1.0005 has 5 significant figures.
  • 300 is ambiguous, use scientific notation $3 \times 10^2$ or $3.00 \times 10^2$.
• Zeros used only for decimal point placement are not significant.
  1. 0.0062 cm has 2 significant figures.
  2. 4.0500 cm has 5 significant figures.

Rules for significant digits:

1. Multiplication/Division: The result has the same number of significant digits as the least accurate factor.
   • Example: $\frac{314.7 \times 45}{20.46} = 693.48 \approx 7.0 \frac{N}{m^2}$. Least significant factor (45) has only two (2) digits.
2. Addition/Subtraction: The result has the same number of decimal places as the term with the fewest decimal places.
   • Example: $9.65 \text{ cm} + 8.4 \text{ cm} - 2.89 \text{ cm} = 15.16 \text{ cm} \approx 15.2 \text{ cm}$

• Area Example: $A = LW = (8.71 \text{ cm})(3.2 \text{ cm}) = 27.872 \text{ cm}^2 \approx 28 \text{ cm}^2$

General Rules to Determine Significant Digits in A Given Number

1. All non-zero numbers are significant.
2. Zeros within a number are always significant.
3. Zeros that do nothing but set the decimal point are not significant. Both 0.000098 and 0.98 contain two significant figures.
4. Zeros that aren’t needed to hold the decimal point are significant. For example, 4.00 has three significant figures.
5. Zeros that follow a number may be significant.

1.3 Vectors: Composition and Resolution

• Scalar: Quantity with magnitude only. Obeys ordinary algebra. Examples: mass, time, volume, speed.
• Vector: Quantity with magnitude and direction. Obeys vector algebra. Examples: displacement, velocity, acceleration, momentum.

1.3.1 Vector Representation

A. Algebraic Method

• Vectors are represented by a letter with an arrow (e.g., $\vec{v}$, $\vec{p}$).
• Magnitude is a positive scalar: $|A|$ or A

B. Geometric Method

• Vectors as straight arrows.
• Zero vector: A point.
• Length represents magnitude.
• Vectors can be parallel transported.
• Addition: $\vec{A} + \vec{B}$ is the arrow from the start of $\vec{A}$ to the end of $\vec{B}$.
• Vectors change if magnitude or direction changes.
• Add/subtract/equate quantities of same units and character.
• Scalar multiplication changes magnitude; negative reverses direction.

1.3.2 Vector Addition

• Resultant vector: Sum of two or more vectors.

A. Graphical Method

• Join head to tail.
• Resultant: Vector from the tail of the first to the head of the last.

B. Parallelogram Law

• Resultant is the diagonal of a parallelogram formed by the two vectors.
• Cosine law: $|R| = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$
• Sine law: $\frac{\sin(\alpha)}{B} = \frac{\sin(\beta)}{A} = \frac{\sin(\theta)}{R}$

1.3.3 Components of Vector

• $A_x = A \cos(\theta)$ (x component)
• $A_y = A \sin(\theta)$ (y component)
• $A = A<em>x + A</em>y$
• Magnitude: $|A| = \sqrt{A<em>x^2 + A</em>y^2}$
• 3D vector: $A = A<em>x + A</em>y + A_z$
• Magnitude: $|A| = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$
• Direction cosines:
  • $\cos(\alpha) = \frac{A_x}{|A|}$
  • $\cos(\beta) = \frac{A_y}{|A|}$
  • $\cos(\gamma) = \frac{A_z}{|A|}$

1.4 Unit Vector

• Vector with magnitude one, dimensionless, indicates direction.
• Denoted with a "hat" (e.g., $\hat{r}$).
• Cartesian coordinate unit vectors:
  • $\hat{i}$ (+x direction)
  • $\hat{j}$ (+y direction)
  • $\hat{k}$ (+z direction)
• Any vector can be expressed using unit vectors.
• $\vec{A} = A<em>x \hat{i} + A</em>y \hat{j} + A_z \hat{k}$

1.4.1 Vector Addition in Unit Vector Notation

• Add corresponding components:
• If $\vec{A} = A<em>x \hat{i} + A</em>y \hat{j} + A<em>z \hat{k}$ and $\vec{B} = B</em>x \hat{i} + B<em>y \hat{j} + B</em>z \hat{k}$, then
• $\vec{A} + \vec{B} = (A<em>x + B</em>x) \hat{i} + (A<em>y + B</em>y) \hat{j} + (A<em>z + B</em>z) \hat{k}$

1.4.2 Finding a Unit Vector

• Unit vector in the direction of $\vec{A}$ is $\hat{r} = \frac{\vec{A}}{|A|}$.
• If $\vec{A} = A<em>x \hat{i} + A</em>y \hat{j} + A_z \hat{k}$, then
• $\hat{r} = \frac{A<em>x}{\sqrt{A</em>x^2 + A<em>y^2 + A</em>z^2}} \hat{i} + \frac{A<em>y}{\sqrt{A</em>x^2 + A<em>y^2 + A</em>z^2}} \hat{j} + \frac{A<em>z}{\sqrt{A</em>x^2 + A<em>y^2 + A</em>z^2}} \hat{k}$

Chapter Summary

• Physical quantity: Quantifiable property.
• Measurement: Comparing quantity to a unit.
• Basic quantities: Independent quantities (length, mass, time).
• Derived quantities: Expressed using basic quantities (area, volume, density).
• Uncertainty: Range of possible values.
• Scalar: Magnitude only.
• Vector: Magnitude and direction.
• Resultant vector: Sum of vectors.
• Unit vector: Magnitude of one, indicates direction.