Work & Energy — Comprehensive Study Notes

Myth of Sisyphus & Conceptual Prelude

• Narrative anchor: Greek myth of Sisyphus → condemned by Zeus to eternally push a boulder up a hill.
  • Each near-success is followed by the rock rolling back; cycle = endless, futile work.
• Physics framing
  • Up-hill push = Sisyphus does work on the boulder → increases its gravitational potential energy (GPE).
  • Rock’s descent = conversion of GPE → kinetic energy (KE).
• Pedagogical purpose: Myth serves as tangible model for:
  1. Definition of work.
  2. Interconversion between KE and PE (mechanical energy exchange).
  3. Preview of mechanical advantage (using ramps/pulleys to mitigate heavy lifting).
• Philosophical nod: Kaplan claims MCAT prep ≠ Sisyphean—progress is retained!

Energy: Broad Definition & Taxonomy

• General definition: Energy is the ability of a system to do work or, more broadly, to make something happen.
• Key forms (with real-world links):
  • Mechanical (motion/acceleration of macroscopic objects).
  • Thermal (random molecular motion; e.g., ice cube melts by heat absorption).
  • Sound (pressure waves in a medium).
  • Light/Radiant (electromagnetic waves).
  • Chemical potential (stored in molecular bonds; released via metabolism).
  • Electrical potential (charge separation, Ch. 5 reference).
  • Nuclear binding (mass-energy release in fissione.g., power plants).

Modes of Energy Transfer

• Only two fundamental mechanisms:
  1. Work (W): force acting through a displacement.
  2. Heat (Q): transfer of thermal energy due to temperature difference.

Kinetic Energy (KE)

• Definition: Energy of motion for an object of mass $m$ moving with speed $v$.
• Formula: $K = \frac{1}{2} m v^{2}$
• Units: Joule (J) $= \text{kg}\,\text{m}^2\,\text{s}^{-2}$ (same for all energy forms).
• Conceptual highlights:
  • KE depends on speed (scalar), not velocity (vector); direction irrelevant.
  • Quadratic relationship: doubling $v$ ⇒ KE ↑ by factor of $4$ (if $m$ constant).
• Cross-chapter link: Dynamic pressure term in Bernoulli’s equation (fluids, Ch. 4).

Worked Example – Frictionless Incline

• Given: $m=15\,\text{kg}$, starts from rest ($v<em>i=0$), bottom speed $v</em>f=7\,\text{m/s}$.
• Top: $K_i=0$ J.
• Bottom: $K<em>f=\tfrac12 m v</em>f^2\approx\tfrac12(15)(7^2)=375\,\text{J}\;(\text{exact }367.5\,\text{J}).$

Potential Energy (PE)

• Concept: Stored energy due to position or intrinsic system properties; has the potential to do work.
• Representative subclasses (MCAT emphasis):
  1. Gravitational PE (GPE)
  2. Elastic PE
  • Others (chemical, electrical, nuclear) acknowledged but treated elsewhere.

Gravitational Potential Energy

• Equation: $U_g = m g h$
  • $h$ = height relative to a chosen datum (zero-PE reference).
• Proportionalities: $U<em>g \propto m, g, h$; tripling any parameter triples $U</em>g$.
• Datum choice strategy: Pick the most convenient reference (floor, tabletop, sea level, etc.).

Worked Example – Cliff Dive

• Data: $m=80\,\text{kg}$, cliff height $10\,\text{m}$, diver ends $2\,\text{m}$ below sea level.
• Top of cliff: $U_{top}=mgh=80(9.8)(10)\approx7.84\times10^{3}\,\text{J}.$
• At –2 m (underwater): $U=-80(9.8)(2)\approx-1.57\times10^{3}\,\text{J}.$
  • Negative sign → below datum (sea level).

Elastic Potential Energy

• Context: Springs/elastic systems displaced from equilibrium.
• Formula: $U_{\text{elastic}} = \frac{1}{2} k x^{2}$
  • $k$ = spring constant (stiffness).
  • $x$ = magnitude of displacement (stretch or compression).
• Structural similarity: Mirrors KE expression; both quadratic in a variable (speed $v$ vs. displacement $x$).
• Caveat: Real springs exhibit internal friction → thermal losses (nonconservative in practice).

Total Mechanical Energy (TME)

• Definition: $E = U + K$ (sum of all PE + KE components considered mechanical).
• Thermodynamic framing: Reflects First Law: energy cannot be created/destroyed, only converted.
  • However, TME ignores non-mechanical forms (e.g., heat due to friction). When those occur, $E$ need not stay constant.

Conservative vs. Nonconservative Forces

Conservative Forces

• Properties:
  • Path-independent work.
  • No net energy loss around a closed loop.
  • Admit well-defined potential energies.
• MCAT-relevant examples: gravitational, electrostatic, (ideal) elastic.
• Tests for conservativeness:
  1. Closed-path criterion: Object returns to start → $\Delta E = 0$ irrespective of trajectory.
  2. Displacement criterion: $\Delta E$ between two points same for any path.

Nonconservative Forces

• Traits:
  • Path-dependent; dissipate mechanical energy (convert to heat, sound, etc.).
  • Examples: kinetic friction, air resistance, viscous drag.
• Energy accounting equation: $W_{\text{noncons}} = \Delta E = \Delta U + \Delta K$
  • Negative $W_{\text{noncons}}$ → energy lost from mechanical system (usually as heat).
  • Longer path ⇒ larger magnitude of dissipation.

Worked Example – Air-Resisted Baseball

• Given: $m=0.25\,\text{kg}, v<em>i=30\,\text{m/s}, v</em>f=27\,\text{m/s}, \;h<em>i=h</em>f\Rightarrow\Delta U=0$.
• Compute:
  $W<em>{air}=\Delta K = \tfrac12 m(v</em>f^2 - v_i^2) = \tfrac12(0.25)(27^2 - 30^2)$
  $=\tfrac18(729 - 900) = -21.4\,\text{J}$ (≈ –20 J in estimate).
• Interpretation: – sign indicates ~21 J of mechanical energy lost to air as heat/sound.

Conservation of Mechanical Energy (CME)

• Ideal statement (no nonconservative forces):
  $\Delta E = \Delta U + \Delta K = 0 \quad \Rightarrow \quad E<em>i = E</em>f$
• Practical statement (nonconservative forces present):
  $E<em>i + W</em>{\text{noncons}} = E_f$
• MCAT test tip: When friction/drag negligible, CME often faster than multi-step kinematics.

Forward Look: Work-Energy Theorem & Mechanical Advantage (teaser)

• Work-Energy Theorem (WET): Net work on a system equals its change in KE (full derivation in later section/problem-solving strategy).
• Mechanical Advantage (MA): Using simple machines (inclined planes, pulleys, levers) to reduce input force required to lift/translate loads.
  • Will explore quantitative MA ratios, efficiency, and energetic trade-offs (upcoming material).

Key Equations – Quick Reference

• Kinetic Energy: $K = \tfrac12 m v^2$.
• Gravitational PE: $U_g = m g h$.
• Elastic PE: $U_e = \tfrac12 k x^2$.
• Total Mechanical Energy: $E = U + K$.
• CME (no dissipation): $\Delta U + \Delta K = 0$.
• Nonconservative work: $W_{\text{noncons}} = \Delta U + \Delta K$.

MCAT Strategy & Conceptual Insights

• Identify whether forces present are conservative; if so, energy-method often speeds problem.
• Draw datum lines clearly to avoid sign mistakes in $h$ for GPE.
• Watch for phrases like frictionless, vacuum, no air resistance ⇒ signals CME applicability.
• Check dimensional consistency; Joule always expressed as $\text{kg}\,\text{m}^2\,\text{s}^{-2}$.
• Be comfortable converting qualitative statements ("doubles speed") into quantitative changes (KE ×4).
• Remember speed vs. velocity: KE disregards direction.

Real-World/Interdisciplinary Connections

• Engineers choose datums strategically in structural analysis (e.g., ground level for skyscrapers).
• Springs in car suspensions: non-ideal → heat buildup after prolonged compression cycles.
• Biomechanics: Gravitational PE ↔ KE interplay in human gait; elastic storage in tendons.
• Environmental science: Energy losses to drag crucial in fuel-efficiency modeling.

Ethical & Philosophical Reflection

• Sisyphus myth illustrates psychological weight of purposeless work; in contrast, energy conservation laws offer predictability and purpose within physical systems.
• Understanding energetics aids ethical technology design (e.g., minimizing waste heat, improving machine efficiency).