Units, Physical Quantities, and Vectors - University Physics Notes

The Nature of Physics

Physics is defined as an experimental science. Within this field, physicists actively seek patterns that serve to relate the various phenomena occurring in nature.
Physical theories are the terms used to describe these identified patterns.
A physical law or principle refers to a theory that has become very well established or is widely used across the scientific community.

Solving Problems in Physics

A standardized four-step process is utilized to solve any physics problem: 1. Identify: This step involves identifying the relevant physical concepts, identifying the target variables (what you need to find), and noting the known quantities that are either explicitly stated or implied within the problem's text. 2. Set Up: In this phase, you choose the specific equations required to solve the problem and create a visual sketch of the situation to aid understanding. 3. Execute: This is the stage where the mathematical calculations are performed ("doing the math"). 4. Evaluate: Finally, you compare your calculated answer against your initial estimates. If there is a discrepancy, you must reconsider your steps and the logic used.

Idealized Models

To simplify the analysis of complex real-world scenarios, physicists use idealized models.
An example of this is the analysis of a baseball in flight; rather than accounting for every possible variable (like wind or ball spin), an idealized model is used to focus on the core mechanics of the motion.

Standards and Units

Physics relies on three fundamental quantities: length, time, and mass.
The International System (SI), known as Système International, is the most globally recognized system of units.
Standard SI units for fundamental quantities are: * Length: Measured in meters ( $\text{m}$ ). * Time: Measured in seconds ( $\text{s}$ ). * Mass: Measured in kilograms ( $\text{kg}$ ).

Unit Prefixes and Examples

Prefixes are employed to create larger or smaller units for fundamental quantities. Common examples and their physical scales include: * Micro- ( $\mu$ ): $1\,\mu\text{m} = 10^{-6}\,\text{m}$ (The approximate size of some bacteria and living cells). * Kilo- ( $k$ ): $1\,\text{km} = 10^{3}\,\text{m}$ (The distance covered in a typical 10-minute walk). * Milli- ( $m$ ): $1\,\text{mg} = 10^{-6}\,\text{kg}$ (The approximate mass of a grain of salt). * Gram ( $g$ ): $1\,\text{g} = 10^{-3}\,\text{kg}$ (The approximate mass of a paper clip). * Nano- ( $n$ ): $1\,\text{ns} = 10^{-9}\,\text{s}$ (The amount of time it takes for light to travel a distance of $0.3\,\text{m}$ ).

Unit Consistency and Conversions

Dimensional consistency is a requirement for all equations. Every term that is added to another or equated must have identical units. This is often described as ensuring you are adding "apples to apples."
Units must be carried through all stages of a calculation to maintain accuracy.
Conversions to standard units are achieved by forming a ratio of the same physical quantity expressed in two different units and using that ratio as a multiplier.
Example calculation: To find the number of seconds in $3\,\text{min}$ , the conversion is written as: $3\,\text{min} = (3\,\text{min}) \times \left( \frac{60\,\text{s}}{1\,\text{min}} \right) = 180\,\text{s}$

Uncertainty and Significant Figures

The number of significant figures in a measured quantity indicates its level of uncertainty.
Multiplication and Division Rule: The final result can have no more significant figures than the factor with the smallest number of significant figures.
Addition and Subtraction Rule: The result's precision is determined by the term that has the fewest digits to the right of the decimal point.
Calculating accurately is vital; as illustrated by historical train mishaps, even a small percentage error can lead to spectacular and disastrous results.

Vectors and Scalars

A scalar quantity is defined as a quantity that can be fully described by a single number.
A vector quantity is defined by having both a magnitude and a specific direction in space.
Notation: In textbook formatting, a vector quantity is represented using boldface italic type with an arrow overhead (e.g., $\mathbf{\vec{A}}$ ).
Magnitude: The magnitude of vector $\mathbf{\vec{A}}$ is written as either $A$ or $|\mathbf{\vec{A}}|$ .

Drawing and Adding Vectors Graphically

Visual Representation: A vector is drawn as a line ending in an arrowhead. The length of the line represents the magnitude, and the line's orientation shows the direction.
Head-to-Tail Method: To add vectors graphically, place the tail of the second vector at the head of the first. The vector sum (resultant), $\mathbf{\vec{C}}$ , extends from the tail of the first vector to the head of the last vector.
Commutative Property: The order of addition does not change the result: $\mathbf{\vec{A}} + \mathbf{\vec{B}} = \mathbf{\vec{B}} + \mathbf{\vec{A}}$ .
Parallelogram Method: Vectors can also be added by placing them tail to tail and constructing a parallelogram; the diagonal starting from the tails represents the sum.
Parallel and Antiparallel Vectors: * If vectors $\mathbf{\vec{A}}$ and $\mathbf{\vec{B}}$ are parallel, the magnitude of the sum is the sum of the magnitudes: $C = A + B$ . * If vectors $\mathbf{\vec{A}}$ and $\mathbf{\vec{B}}$ are antiparallel, the magnitude of the sum is the absolute difference of the magnitudes: $C = |A - B|$ .
Multiple Vectors: The head-to-tail method remains valid when adding more than two vectors, regardless of the order in which they are added.

Vector Subtraction

Subtracting vector $\mathbf{\vec{B}}$ from $\mathbf{\vec{A}}$ is defined as adding the negative of vector $\mathbf{\vec{B}}$ to $\mathbf{\vec{A}}$ : $\mathbf{\vec{A}} - \mathbf{\vec{B}} = \mathbf{\vec{A}} + (-\mathbf{\vec{B}})$ .
Visualizations: 1. With $\mathbf{\vec{A}}$ and $\mathbf{\vec{-B}}$ placed head-to-tail, $\mathbf{\vec{A}} - \mathbf{\vec{B}}$ is the vector from the tail of $\mathbf{\vec{A}}$ to the head of $\mathbf{\vec{-B}}$ . 2. If original vectors $\mathbf{\vec{A}}$ and $\mathbf{\vec{B}}$ are placed tail-to-tail, $\mathbf{\vec{A}} - \mathbf{\vec{B}}$ is the vector directed from the head of $\mathbf{\vec{B}}$ to the head of $\mathbf{\vec{A}}$ .

Multiplication of a Vector by a Scalar

If $c$ is a scalar and $\mathbf{\vec{A}}$ is a vector, the product has a magnitude of $|c|A$ .
A positive scalar (c > 0) maintains the original direction of the vector.
A negative scalar (c < 0) reverses the direction of the vector.

Vector Components and Calculations

Because graphical addition has limited accuracy, vector components provide a more precise and general method.
Any vector can be resolved into an x-component ( $A_x$ ) and a y-component ( $A_y$ ). These components can be positive or negative depending on their direction relative to the axes.
Finding Magnitude and Direction from Components: * Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ * Direction: $\tan(\theta) = \frac{A_y}{A_x}$
Adding Vectors using Components: To find the components of a resultant vector $\mathbf{\vec{R}}$ from a set of vectors ( $\mathbf{\vec{A}}$ , $\mathbf{\vec{B}}$ , $\mathbf{\vec{C}}$ , etc.): * $R_x = A_x + B_x + C_x + \dots$ * $R_y = A_y + B_y + C_y + \dots$
Example Case: A cross-country skier moving in vectors at right angles results in a displacement of $2.24\,\text{km}$ at an angle of $63.4^{\circ}$ East of North.

Unit Vectors

A unit vector has a magnitude of exactly $1$ and carries no units.
Standard unit vectors define directions along the axes: * $\hat{i}$ points in the $+x$ -direction. * $\hat{j}$ points in the $+y$ -direction. * $\hat{k}$ points in the $+z$ -direction.
Any vector can be expressed as: $\mathbf{\vec{A}} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ .

The Scalar (Dot) Product

The scalar product of vectors $\mathbf{\vec{A}}$ and $\mathbf{\vec{B}}$ is defined as: $\mathbf{\vec{A}} \cdot \mathbf{\vec{B}} = AB \cos(\phi) = |\mathbf{\vec{A}}||\mathbf{\vec{B}}| \cos(\phi)$
Where $\phi$ is the angle between the two vectors when they are placed tail-to-tail.
Interpretations: * $\mathbf{\vec{A}} \cdot \mathbf{\vec{B}}$ equals the magnitude of $\mathbf{\vec{A}}$ multiplied by the component of $\mathbf{\vec{B}}$ in the direction of $\mathbf{\vec{A}}$ : $A(B \cos(\phi))$ * $\mathbf{\vec{A}} \cdot \mathbf{\vec{B}}$ equals the magnitude of $\mathbf{\vec{B}}$ multiplied by the component of $\mathbf{\vec{A}}$ in the direction of $\mathbf{\vec{B}}$ : $B(A \cos(\phi))$
Characteristics: The scalar product can be positive, negative, or zero based on the value of the angle $\phi$ .
Component Calculation: $\mathbf{\vec{A}} \cdot \mathbf{\vec{B}} = A_x B_x + A_y B_y + A_z B_z$ .

The Vector (Cross) Product

The vector product of two vectors results in a new vector $\mathbf{\vec{C}} = \mathbf{\vec{A}} \times \mathbf{\vec{B}}$ .
Magnitude: The magnitude is given by: $C = AB \sin(\phi)$
Direction: The direction of the resulting vector is determined by the Right-Hand Rule.
Anticommutative Property: Unlike scalar multiplication, the order of the cross product matters: $\mathbf{\vec{A}} \times \mathbf{\vec{B}} = -(\mathbf{\vec{B}} \times \mathbf{\vec{A}})$ .