Fundamental Concepts in Physics: Measurement, Vectors, and Kinematics

1.3 Standards and Units

Introduction to Measurement in Physics

Physics relies on precise measurement, which requires standardized units and systems. Understanding these standards is essential for consistency and communication in scientific work.

• Physical Quantity: Any property that can be measured, such as length, mass, or time.

• Standard Units: Agreed-upon units for measurement, such as the meter (m) for length, kilogram (kg) for mass, and second (s) for time.

• SI System: The International System of Units (SI) is the most widely used system in science.

Example: The speed of light is measured as $c = 2.998 \times 10^8$ m/s.

1.5 Uncertainty and Significant Figures

Precision and Reporting in Measurements

All measurements have some degree of uncertainty. Significant figures communicate the precision of a measurement.

• Uncertainty: The range within which the true value is expected to lie.

• Significant Figures: Digits in a measurement that are known reliably plus one estimated digit.

• Rules: When performing calculations, the result should not be more precise than the least precise measurement.

Example: If you measure a length as 12.3 cm, the uncertainty might be ±0.1 cm, and the value has three significant figures.

Additional info: Uncertainty can be expressed as absolute (e.g., ±0.1 cm) or relative (e.g., 0.8%).

1.7 Vectors and Vector Addition

Understanding Directional Quantities

Many physical quantities have both magnitude and direction, known as vectors. Adding vectors requires considering both aspects.

• Vector: A quantity with both magnitude and direction (e.g., displacement, velocity).

• Scalar: A quantity with only magnitude (e.g., mass, temperature).

• Vector Addition: Vectors are added using the parallelogram rule or triangle rule.

Equation:

$\vec{A} + \vec{B} = \vec{C}$

Example: Walking 3 m east and then 4 m north results in a displacement vector found by vector addition.

1.8 Components of a Vector

Breaking Vectors into Parts

Vectors can be resolved into components along chosen axes, simplifying calculations and analysis.

• Component: The projection of a vector along an axis (usually x and y).

• Resolution: Any vector $\vec{A}$ can be written as $A_x$ and $A_y$.

Equation:

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

Example: A force of 10 N at 30° above the horizontal has components $A_x = 10 \cos 30°$, $A_y = 10 \sin 30°$.

1.9 Unit Vectors

Expressing Direction in Vector Notation

Unit vectors are used to specify direction and simplify vector expressions.

• Unit Vector: A vector with magnitude 1, indicating direction (e.g., $\hat{i}$, $\hat{j}$, $\hat{k}$ for x, y, z axes).

• Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

Example: A velocity vector $\vec{v} = 3 \hat{i} + 4 \hat{j}$ m/s.

1.10 Products of Vectors

Multiplying Vectors: Dot and Cross Products

Vectors can be multiplied in two main ways: dot product (scalar) and cross product (vector).

• Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta$ (results in a scalar)

• Cross Product: $\vec{A} \times \vec{B} = AB \sin \theta \hat{n}$ (results in a vector perpendicular to both)

Example: Work done $W = \vec{F} \cdot \vec{d}$ uses the dot product.

2.1 Displacement, Time, and Average Velocity

Describing Motion in One Dimension

Kinematics studies the motion of objects. Displacement, time, and velocity are fundamental concepts.

• Displacement ($\Delta x$): Change in position, a vector quantity.

• Time ($t$): Duration over which motion occurs.

• Average Velocity ($v_{avg}$): Displacement divided by time interval.

Equation:

$v_{avg} = \frac{\Delta x}{\Delta t}$

Example: If a car moves 100 m east in 5 s, $v_{avg} = 20$ m/s east.

2.2 Instantaneous Velocity

Velocity at a Specific Moment

Instantaneous velocity is the rate of change of position at a particular instant.

• Definition: The derivative of position with respect to time.

Equation:

$v = \frac{dx}{dt}$

Example: The speedometer of a car shows the instantaneous velocity.

2.3 Average and Instantaneous Acceleration

Describing Changes in Velocity

Acceleration measures how quickly velocity changes. It can be average over a time interval or instantaneous at a moment.

• Average Acceleration ($a_{avg}$): Change in velocity divided by time interval.

• Instantaneous Acceleration ($a$): The derivative of velocity with respect to time.

Equations:

$a_{avg} = \frac{\Delta v}{\Delta t}$ $a = \frac{dv}{dt}$

Example: If a car's velocity increases from 0 to 20 m/s in 4 s, $a_{avg} = 5$ m/s2.