Definitions

• Work (W): Energy transferred by a force acting over a distance, is scalar and measured in Joules
  • W = F · d = Fd cos θ
    • Is a dot product
    • Can equal change in energy(ΔE) depending on context
• Energy(E): The capacity to do work, is a scalar and measured in Joules

Dot Product

The dot product (·) is a scalar product of two vectors:

• A · B = |A||B| cos θ

This explains why work is maximum when force and displacement are parallel (θ = 0°) and zero when perpendicular (θ = 90°).

Kinetic and Potential Energy

• Kinetic Energy (KE): Energy of motion (if an object is moving it has kinetic energy)
  • KE = ½mv²
• Gravitational Potential Energy (PE): Energy due to position in a gravitational field
  • UG = mgh
• Elastic Potential Energy: Energy stored in springs or elastic materials
  • US = ½kx²

Conservation of Energy

Energy cannot be created or destroyed, only transformed from one form to another. In a closed system:

• Etotal = KE + PE = constant

Work-Energy Theorem

The net work done on an object equals its change in kinetic energy:

• Wnet = ΔKE = KEf - KEi

Conservative vs Non-Conservative Forces

• Conservative Forces: Work done is path-independent; energy is conserved (gravity, spring force, electric force)
• Non-Conservative Forces: Work done depends on path; energy is dissipated (friction, air resistance)
  • Friction: f = μN (where μ is coefficient of friction, N is normal force)
  • Air Resistance/Drag: Opposes motion through a fluid, depends on speed and shape