Physics Fundamentals: Units, Vectors, Kinematics, and Newton's Laws – Practice and Concepts

Units, Conversions, and Dimensional Analysis

SI Units and Volume Conversions

Physics relies on the International System of Units (SI) for consistency in measurements. Understanding how to convert between units is essential for solving problems.

• Volume: The SI unit for volume is the cubic meter (m3), but liters (L) and cubic centimeters (cm3) are also commonly used.

• Conversions:

  • 1 L = 1,000 cm3

  • 1 cm3 = 1 mL

  • 1 m3 = 1,000,000 cm3

• Example: To convert 1 L to cm3: $1\ \text{L} = 1,000\ \text{cm}^3$

Speed and Unit Conversion

• Speed: The SI unit for speed is meters per second (m/s).

• Conversion Example: To convert miles per hour (mph) to m/s:

  • 1 mile = 1,609 m

  • 1 hour = 3,600 s

  • Formula: $\text{Speed (m/s)} = \text{Speed (mph)} \times \frac{1,609}{3,600}$

Area and Volume Calculations

• Area: For rectangles, $A = \text{length} \times \text{width}$

• Volume of a Cylinder: $V = \pi r^2 h$ where $r$ is the radius and $h$ is the height.

• Example: For a cylinder with $r = 4.17\ \text{cm}$ and $h = 12\text{cm}$:

  • $V = \pi (4.17)^2 \times 12$

Vectors and Their Properties

Vector Addition and Representation

Vectors are quantities with both magnitude and direction. They can be added graphically or analytically.

• Graphical Addition:

  • Tail-to-tail

  • Head-to-tail (most common method)

  • Head-to-head

  • Parallel

• Components of a Vector: For a vector $\vec{A}$ at angle $\theta$:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Example: For $A = 5.0\ \text{cm}$, $\theta = 30^\circ$:

  • $A_x = 5.0 \cos 30^\circ = 4.33\ \text{cm}$

  • $A_y = 5.0 \sin 30^\circ = 2.5\ \text{cm}$

Kinematics: Motion in One and Two Dimensions

Speed, Velocity, and Acceleration

• Speed: Scalar quantity; rate of distance traveled.

• Velocity: Vector quantity; rate of displacement.

• Acceleration: Rate of change of velocity.

  • $a = \frac{\Delta v}{\Delta t}$

• Example: If a car accelerates from rest at $5.0 \times 10^3\ \text{m/s}^2$ for $2.0 \mu s$:

  • $v = a t = 5.0 \times 10^3 \times 2.0 \times 10^{-6} = 1.0 \times 10^{-2}\ \text{m/s}$

Graphical Analysis of Motion

• Velocity vs. Time Graph: The slope represents acceleration.

• Types of Acceleration:

  • Instantaneous: At a specific moment.

  • Average: Over a time interval.

  • Minimum/Maximum: Lowest/highest value on the graph.

Projectile Motion and Relative Velocity

• Projectile Motion: Objects launched into the air follow a parabolic path under gravity.

• Key Equations:

  • Horizontal: $x = v_{0x} t$

  • Vertical: $y = v_{0y} t - \frac{1}{2} g t^2$

• Relative Velocity: The velocity of an object as observed from a particular frame of reference.

• Example: A boat crossing a river with current; use vector addition to find resultant velocity and time to cross.

Newton's Laws of Motion and Forces

Types of Forces

• Force: A push or pull on an object; SI unit is the Newton (N).

• Types:

  • Normal Force: Perpendicular to the surface.

  • Tension: Force transmitted through a string or rope.

  • Friction: Opposes motion between surfaces.

  • Restoring Force: Acts to return a system to equilibrium.

Newton's Second Law

• Statement: The net force on an object equals its mass times its acceleration.

• Equation: $F = m a$

• Example: For a mass $m = 100\ \text{kg}$ pulled with $F = 500\ \text{N}$ at $30^\circ$:

  • Horizontal component: $F_x = F \cos 30^\circ$

  • $a = \frac{F_x}{m}$

Inclined Planes and Friction

• Inclined Plane: An object on a slope experiences gravitational, normal, and frictional forces.

• Frictional Force: $f = \mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal force.

• Minimum Force to Move an Object Up an Incline:

  • Consider both gravity and friction.

  • Resolve applied force into components parallel and perpendicular to the incline.

Sample Table: Types of Forces

Force Type	Description	Direction
Normal	Perpendicular contact force from a surface	Perpendicular to surface
Tension	Force transmitted by a string or rope	Along the string/rope
Friction	Opposes relative motion between surfaces	Parallel to surface, opposite to motion
Restoring	Returns system to equilibrium (e.g., spring)	Toward equilibrium position

Problem-Solving Tips

• Read all practice problems carefully and identify knowns and unknowns.

• Draw diagrams for vector and motion problems.

• Convert all quantities to SI units before calculations.

• Use appropriate formulas and check units in your answers.

• Review key concepts and practice regularly for mastery.

Additional info: Some explanations and formulas were expanded for clarity and completeness based on standard introductory physics curriculum.