Chapter 1: Models, Measurements & Vectors – Study Notes

Chapter 1: Models, Measurements & Vectors

Mathematical Tools Required

Physics relies on mathematical concepts to describe and analyze physical phenomena. In this chapter, students are expected to use:

• Algebra: Manipulating equations and solving for unknowns.

• Trigonometry: Calculating angles and sides in right triangles, essential for resolving vectors.

• Geometry: Understanding shapes, distances, and spatial relationships.

Units & Dimensions

Physical quantities must be measured and expressed using standardized units and dimensions.

• Dimension: A physical quantity (e.g., length, mass, time).

• Unit: A standard way to measure a physical quantity (e.g., meter for length, kilogram for mass).

Precision & Significant Figures

• When performing calculations, keep one more digit than needed for your final answer to maintain accuracy.

• Significant figures reflect the precision of a measurement.

Vectors and Scalars

Many physical quantities are described by both magnitude and direction, while others have only magnitude.

• Vector: A mathematical object with both magnitude (size) and direction. Examples: velocity, acceleration, force, displacement.

• Scalar: A mathematical object with only magnitude. Examples: speed, temperature, pressure, distance.

• Notation: Vectors are often denoted with an arrow ($\vec{A}$) or boldface ($\mathbf{A}$). The magnitude of a vector $\vec{A}$ is written as $|\vec{A}|$.

Vector Magnitude

• The magnitude of a vector $\vec{A}$ is $|\vec{A}| = A$.

Resolving Vectors into Components

Vectors can be decomposed into components along coordinate axes, which simplifies calculations and analysis.

• Each component represents the "amount" of the vector in a particular direction (e.g., x or y).

• The components and the vector form a right triangle.

• Use trigonometry to determine components:

$A_x = A \cos \theta$ $A_y = A \sin \theta$

• $A_x$ and $A_y$ are the scalar components of vector $\vec{A}$.

Example 1: Finding Components

A displacement vector $\vec{d}$ in the xy-plane is 15 m long and directed 250° clockwise from the +y axis. Find the components of $\vec{d}$.

• Solution: $d_x = -14.1$ m, $d_y = -5.13$ m

Example 2: Magnitude and Direction from Components

The x-component of $\vec{A}$ is –25 m and the y-component is +40 m. Find the magnitude and direction of $\vec{A}$.

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Direction: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Solution: $A = 47.2$ m, $\theta = 122^\circ$ counterclockwise from +x axis

Vector Addition and Subtraction

Vectors can be added or subtracted both graphically and analytically.

Graphical Addition

• Place vectors tip-to-tail; the resultant vector is drawn from the tail of the first to the tip of the last.

• For vectors $\vec{A}$ and $\vec{B}$: $\vec{A} + \vec{B} = \vec{C}$

Analytical Addition/Subtraction

• Resolve each vector into components.

• Add or subtract corresponding components:

$\vec{a} + \vec{b} = \vec{c}$ $a_x + b_x = c_x$ $a_y + b_y = c_y$

• To subtract vectors: $-\vec{b} = -b_x, -b_y$ (flip the sign of each component).

• A minus sign reverses the direction but does not change the magnitude.

Example Problem 1

Guangdang needs to travel 1.2 m northeast to find the missing arm of her sweater. After walking 0.6 m east, calculate:

1. The x- and y-components of the displacement vector she should have traveled.

2. The distance and direction she now needs to walk to find her sweater.

• Solution: (a) 0.85 m each direction; (b) 0.88 m, 73.7° N of E

Example Problem 2

Two vectors $\vec{a}$ and $\vec{b}$ have equal magnitudes of 10.0 m and angles $\theta_1 = 30^\circ$, $\theta_2 = 105^\circ$. Find:

1. The x- and y-components of $\vec{c} = \vec{a} + \vec{b}$

2. The magnitude of $\vec{c}$

3. The angle $\vec{c}$ makes with the +x axis

• Solution: $|\vec{c}| = 12.2$ m, $\theta_c = 82.5^\circ$

Summary Table: Scalars vs. Vectors

Quantity Type	Definition	Examples
Scalar	Has magnitude only	Speed, temperature, pressure, distance
Vector	Has magnitude and direction	Velocity, acceleration, force, displacement

Additional info: These notes expand on the brief points in the slides and handwritten notes, providing full definitions, formulas, and worked examples for clarity and completeness.