Work, Energy & Power – Comprehensive Notes

Work

• Definition
  • Work is done when a body is displaced under the action of a force.
  • Vector definition: work is the scalar (dot) product of force and displacement.
  • $W = \vec F \cdot \vec d = F d \cos \theta$
  • $\theta$ = angle between $\vec F$ and $\vec d$.
• Units and nature
  • Scalar quantity (direction-independent).
  • SI unit: joule (J) $1\,\text{J} = 1\,\text{N\,m}$.
• Categories of work
  • Positive work
  • Condition: the component of force is parallel to displacement 0^\circ \le \theta < 90^\circ.
  • Maximum value when $\theta = 0^\circ$ ⇒ $W_{\text{max}} = Fd$.
  • Example: Lifting a body upward—the upward lifting force does positive work.
  • Negative work
  • Force component is opposite to displacement 90^\circ < \theta \le 180^\circ.
  • Minimum (largest negative) when $\theta = 180^\circ$ ⇒ $W_{\text{min}} = -Fd$.
  • Examples:
    • While lifting a body, gravity (downward) does negative work.
    • Friction on a body sliding over a rough surface opposes motion, doing negative work.
  • Zero work
  • Situations that yield $W = 0$:
    1. Force ⟂ displacement ($\theta = 90^\circ$)
    • Coolie walking horizontally with load on head (no vertical displacement against gravity).
    • Centripetal force in uniform circular motion (always radially inward, perpendicular to tangential displacement).
    1. Displacement $d = 0$ while force acts
    • Pushing an immovable wall or stone.
    • Weight-lifter holding a weight stationary overhead.
    1. Force $F = 0$ while displacement occurs
    • Motion of an isolated body in free space (no external forces).

Energy

• Definition: capacity of a body to do work; numerically equal to total work it can perform.
• Scalar quantity.
• Units
  • SI: joule (J); CGS: erg.
  • Practical/derived: electron-volt, kilowatt-hour, calorie.
  • $1\,\text{J} = 10^7\,\text{erg}$
  • $1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$
  • $1\,\text{kWh} = 3.6 \times 10^{6}\,\text{J}$
  • $1\,\text{cal} = 4.18\,\text{J}$
• Principal forms (illustrative not exhaustive)
  • Mechanical (kinetic & potential)
  • Chemical, electrical, magnetic, nuclear, sound, light, heat, etc.
• Transformations (illustrative chain of real-world examples)
  • Heat engine: heat ⇒ mechanical.
  • Electric bulb: electrical ⇒ light.
  • Combustion of coal/oil: chemical ⇒ heat.
  • Solar cell: solar ⇒ electrical.
  • Playing sitar: mechanical ⇒ sound.
  • Microphone: sound ⇒ electrical; loudspeaker: electrical ⇒ sound.
  • Battery (during discharge): chemical ⇒ electrical/ mechanical (in special cells).
  • Electric motor: electrical ⇒ mechanical.

Kinetic Energy (KE)

• Definition: energy possessed by a body by virtue of its motion.
• Mathematical expression
  • For mass $m$ moving with velocity $v$:
    $KE = \frac{1}{2} m v^2$
  • In terms of linear momentum $p = m v$:
    $KE = \frac{p^2}{2m}$
• Scaling rules
  • Doubling $v$ (or $p$) increases $KE$ by factor 4.
  • For two bodies with equal momentum ($p<em>1 = p</em>2$):
    $\frac{E<em>1}{E</em>2} = \frac{m<em>2}{m</em>1}$ (lighter mass has greater KE).
  • For equal kinetic energies ($E<em>1 = E</em>2$):
    $\frac{p<em>1}{p</em>2} = \sqrt{\tfrac{m<em>1}{m</em>2}}$.
• Everyday examples
  • Flowing water → drives water mills.
  • Moving vehicles, wind (driving windmills), hammer striking nail, bullet penetrating target.

Potential Energy (PE)

• Definition: energy possessed due to position in a field of conservative forces.
• Common types
  • Gravitational, elastic, electric.
• Near-Earth gravitational PE
  $PE = m g h$
  where $h$ = height above reference level.
• Absolute gravitational potential energy: work done in bringing a body from infinity to a point.

Power

• Concept: rate of doing work / transferring energy.
• Mathematical forms
  • Average power: $P = \frac{W}{t}$ (watts).
  • Instantaneous (vector) form using velocity:
    $P = \vec F \cdot \vec v = F v \cos \theta$.
• Nature: scalar (follows dot-product rules).
• Units & conversions
  • SI: watt $\bigl(1\,\text{W} = 1\,\text{J s}^{-1} = 10^7\,\text{erg s}^{-1}\bigr)$.
  • Practical: kilowatt, megawatt, horsepower.
  • $1\,\text{kW} = 10^{3}\,\text{W}$
  • $1\,\text{MW} = 10^{6}\,\text{W}$
  • $1\,\text{hp} = 746\,\text{W}$
• Comparative rule (equal work):
  • $P \propto \frac{1}{t}$ → the faster the work, the higher the power.

Conservation of Energy

• Statement: The total energy (including mass-energy) of the universe remains constant; it can only change form.
• Illustrations
  • Free-fall: gravitational PE converts to KE.
  • Simple pendulum: periodic interchange between KE and PE.
  • Mass-spring oscillations: elastic PE ⇌ KE.
• Diagnostic principle: any process that appears to violate energy conservation cannot occur in reality.