Introduction to Physics with Calculus

Physics with Calculus is a foundational course for scientists and engineers, focusing on the mathematical description of the physical world. This study guide covers the essential concepts from the nature of physics, mathematical tools, kinematics, vectors, and the basics of forces and motion.

Math Review and Scientific Approach

Key Concepts

• Physics is the study of the fundamental laws of nature, emphasizing relationships and patterns over rote memorization of facts.

• Mathematics is a tool in physics, used to describe and analyze patterns in nature.

• Problem Solving in physics involves modeling, visualizing, solving, and reviewing results for reasonableness and correct units.

Units, Measurement, and Significant Figures

SI Units

• Length: meter (m)

• Mass: kilogram (kg)

• Time: second (s)

Significant Figures

• Reflect the precision of measurements.

• When multiplying/dividing, the result should have as many significant figures as the least precise measurement.

• When adding/subtracting, the result should have as many decimal places as the least precise measurement.

Kinematics: Describing Motion

Types of Motion

• Translational Motion: Linear, circular, and projectile motion (object moves through space).

• Rotational Motion: Object rotates about an axis (studied later).

Key Terms

• Position (ru): Location of an object in space, often given as coordinates or a vector.

• Displacement (Δru): Change in position, a vector from initial to final position.

• Velocity (vu): Rate of change of position; a vector with both magnitude (speed) and direction.

• Acceleration (au): Rate of change of velocity; can change speed, direction, or both.

Equations

Motion Diagrams

• Visual tool showing position, velocity, and acceleration at successive time intervals.

• Displacement vectors connect positions; velocity vectors are tangent to the path; acceleration vectors show change in velocity.

Vectors and Vector Operations

Definitions

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

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

Vector Addition and Components

• Vectors are added tip-to-tail or using the parallelogram rule.

• Components: ,

• Magnitude:

• Direction:

Unit Vectors

• Define directions in space: (x-direction), (y-direction).

• Any vector:

1D and 2D Kinematics

1D Motion with Constant Acceleration

Projectile Motion (2D)

• Horizontal motion:

• Vertical motion:

• Range (level ground):

• Time of flight:

Forces and Newton's Laws

What is a Force?

• A force is a push or pull exerted on an object by an agent.

• Forces are vectors: they have both magnitude and direction.

• Types: Contact forces (require physical contact) and long-range forces (act at a distance, e.g., gravity).

Common Forces

• Gravity (FG): Always points downward, magnitude .

• Spring Force (FSp): Pushes or pulls depending on compression or extension.

• Tension (T): Force exerted by a string or rope, always directed along the string.

Free-Body Diagrams

• Visual representation of all forces acting on a particle.

• Forces are drawn as arrows with tails on the object.

Newton's Second Law

• The net force on an object causes it to accelerate:

• All forces acting on an object combine as vectors to form the net force.

Tables and Data

Common Prefixes

Images

Summary

• Physics with Calculus uses mathematical models to describe and predict motion.

• Vectors are essential for representing quantities with direction and magnitude.

• Kinematics describes motion; dynamics (forces) explains why motion changes.

• Newton's laws form the foundation for understanding the relationship between force and motion.