Introduction to Physics and Vectors – PHY 121 Study Notes

Lecture 1: Introduction and Vectors Review

Course Overview

This course, PHY 121, introduces students to the foundational principles of physics, focusing on mechanics and vectors. The course aims to develop the ability to apply a small set of fundamental physical principles to a wide variety of physical situations, emphasizing modeling, technical language, and problem-solving skills.

• Instructor: Dr. Douglas Shepherd, Assistant Professor of Physics, Center for Biological Physics faculty, Neuroscience program graduate faculty.

• Course Format: In-person lectures and recitations, with a mix of lectures, example problems, and group practice exercises.

• Digital Tools: Canvas for course materials and communication, Mastering Physics for homework and practice problems, CampusWire for announcements and questions.

Course Goals

• Application of Principles: Students will learn to apply fundamental physical principles to diverse physical systems.

• Modeling: Emphasis on making approximations and idealizations to simplify complex systems for analysis.

• Technical Communication: Develop proficiency in reading, writing, and speaking the technical language of Newtonian mechanics.

Grading Scheme

The course uses a weighted grading system based on exams, homework, and quizzes. The lowest scores in homework and quizzes are dropped to accommodate student learning curves.

Course Work	Weight
Exams (3 total)	25% each (75% total)
Homework (lowest 2 dropped)	15%
Reading Quiz (lowest 4 dropped)	5%
Recitation Quiz (lowest 2 dropped)	5%

Percentage	Grade
>=96	A+
90-95.9	A
85-89.9	A-
80-84.9	B+
75-79.9	B
70-74.9	B-
65-69.9	C+
60-64.9	C
50-59.9	D
0-49.9	E

Vectors in Physics

Definition and Importance

Vectors are quantities that have both magnitude and direction. They are essential in physics for describing quantities such as displacement, velocity, acceleration, and force.

• Scalar vs. Vector: Scalars have only magnitude (e.g., mass, temperature), while vectors have both magnitude and direction (e.g., velocity, force).

• Notation: Vectors are often denoted with an arrow above the symbol (e.g., $\vec{A}$) or in bold (e.g., A).

Vector Representation

• Graphical Representation: Vectors are represented as arrows; the length indicates magnitude, and the arrow points in the direction.

• Component Form: Any vector in two or three dimensions can be broken into components along the coordinate axes.

$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

• Magnitude: The magnitude of a vector $\vec{A}$ is given by: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Vector Addition and Subtraction

• Tip-to-Tail Method: Place the tail of one vector at the tip of another to find the resultant vector.

• Component Method: Add or subtract corresponding components: $\vec{C} = \vec{A} + \vec{B} \implies C_x = A_x + B_x,\; C_y = A_y + B_y,\; C_z = A_z + B_z$

Multiplication of Vectors

• Dot Product (Scalar Product): $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

• Cross Product (Vector Product): $\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \; \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$.

Applications of Vectors in Physics

• Displacement: The change in position of an object, represented as a vector from the initial to the final position.

• Velocity: The rate of change of displacement with respect to time, also a vector.

• Force: A vector quantity that causes acceleration according to Newton's Second Law.

Example: Calculating the Resultant Displacement

If a student walks 3 meters east and then 4 meters north, the resultant displacement is:

$|\vec{d}| = \sqrt{3^2 + 4^2} = 5$ meters

The direction can be found using trigonometry:

$\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.1^\circ$ north of east

Course Logistics and Expectations

Lecture and Recitation Format

• Lectures will include theoretical explanations, example problems, and group exercises.

• Recitations will begin with a quiz and focus on collaborative problem-solving.

• All course materials, announcements, and assignments are managed through Canvas and Mastering Physics.

Student Preparation

• Students are expected to have some prior exposure to physics and vectors (e.g., AP Physics).

• Active participation in lectures and recitations is encouraged for mastering the material.

Introduction to Biological Physics Context

Micro-organism Propulsion

Physics principles are applied to understand how micro-organisms, such as E. coli, move in their environments. These organisms can move at speeds up to 15 times their body length per second, with flagella rotating at extremely high speeds (up to 18,000 RPM).

• Applications: Understanding propulsion, chemotaxis (movement in response to chemical gradients), and biofilm formation.

• Comparison: The power-to-weight ratio of E. coli flagella can be compared to that of high-performance animals and machines.

Additional info: These examples illustrate the relevance of vectors and mechanics in biological systems, connecting course concepts to real-world phenomena.

Summary

• This course will build a foundation in vectors and mechanics, essential for further study in physics and related fields.

• Students will learn to model, analyze, and communicate about physical systems using the language and tools of physics.