Chapter: Work, Energy and Power

• Introduction

  • Definition of Work, Energy, and Power in Physics
  • Scalar (Dot) Product of Vectors
    • Formula: $\vec{A} \cdot \vec{B} = AB \cos\theta$
    • Properties: Commutative, Distributive

• Work ($W$)

  • Scientific Definition
    • Force ($F$) $\times$ Displacement ($s$) $\times \cos\theta$
    • SI Unit: Joule (J) or $\text{N}\cdot\text{m}$
    • Dimension: $[ML^2T^{-2}]$
  • Nature of Work
    • Positive Work: $\theta < 90^\circ$ (e.g., pulling a toy)
    • Negative Work: $\theta > 90^\circ$ (e.g., friction)
    • Zero Work: $\theta = 90^\circ$ (e.g., carrying load horizontally)
  • Work-Energy Theorem
    • Net work done = Change in Kinetic Energy ($W = \Delta K$)
  • Work Done by a Variable Force
    • Area under Force-Displacement graph
    • Integral form: $W = \int_{x_i}^{x_f} F(x) dx$

• Energy ($E$)

  • Definition: Capacity to do work
  • SI Unit: Joule (J)
  • Kinetic Energy ($K$)
    • Energy by virtue of motion
    • Formula: $K = \frac{1}{2}mv^2$
    • Relation with momentum: $K = \frac{p^2}{2m}$
  • Potential Energy ($V$)
    • Energy by virtue of position or configuration
    • Gravitational Potential Energy: $V = mgh$
    • Elastic Potential Energy (Spring): $V = \frac{1}{2}kx^2$
  • Conservative vs. Non-Conservative Forces
    • Conservative Force: Work depends only on initial/final positions (e.g., Gravity, Spring force); Path independent; Total mechanical energy conserved.
    • Non-Conservative Force: Work depends on the path (e.g., Friction, Viscosity); Energy is dissipated as heat.
  • Conservation of Mechanical Energy
    • $E = K + V = \text{constant}$ (under conservative forces)

• Collisions

  • Law of Conservation of Momentum (applies to all collisions)
  • Elastic Collision
    • Conservation of both Momentum and Kinetic Energy
    • Coefficient of restitution ($e = 1$)
    • Example: Billiard balls
  • Inelastic Collision
    • Conservation of Momentum only; Kinetic Energy is not conserved
    • Maximum loss of KE in Perfectly Inelastic Collision ($e = 0$, bodies stick together)
  • Collisions in One and Two Dimensions

• Power ($P$)

  • Definition: Rate of doing work
  • Formula: $P = \frac{W}{t}$ or $P = \vec{F} \cdot \vec{v}$
  • SI Unit: Watt (W) or $\text{J/s}$
  • Practical Unit: Horsepower (hp) ($1 \text{ hp} = 746 \text{ W}$)
  • Commercial Unit of Energy: Kilowatt-hour (kWh) ($1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$)