Energy & Work–Energy Principles – Comprehensive Study Notes

Energy: Core Definition & Fundamental Properties

• Scalar physical quantity defined operationally as “the ability to do work”.

• SI unit: Joule (J)
  • 1\;\text{J}=1\;\text{kg}\,\text{m}^2/\text{s}^2

• Exists in multiple interchangeable “flavours,” but total amount is governed by an overarching universal law (Conservation of Energy).

Universal Law – Conservation of Energy

• Statement: There is a certain numerical quantity, energy, that does not change in any physical process.

• Experimental status: absolutely no known exception; applies to all natural phenomena.

• Applies strictly to an isolated system (no energy exchange with surroundings). The universe itself is treated as the ultimate isolated system → total energy of the universe is constant.

• Tells why processes happen (constraint), not how they occur (mechanism).

• Within a system, energy can transform among forms (kinetic, potential, thermal, chemical, nuclear, electrical, magnetic, electromagnetic, etc.) without altering the overall sum.

Forms / “Flavours” of Energy

• Kinetic Energy (motion)

• Gravitational Potential Energy (position in a gravitational field)

• Elastic / Spring Potential Energy (deformation of elastic media)

• Thermal Energy (random microscopic kinetic + potential)

• Sound, Chemical, Nuclear, Electrical, Magnetic, Electromagnetic (light) energy, etc.

Kinetic Energy (KE)

• Energy possessed by an object due to motion.

• Quantitative expression:
  K = \frac12 m v^2
  • Directly proportional to mass m and to the square of speed v.

• Unit: Joule.

• Doubling speed quadruples KE; tripling speed multiplies KE by 9 (Question 3 scenario).

• Concept checks:
  • “By what factor does KE change if speed is tripled?” → Factor 9.

Work & the Work–Kinetic-Energy Theorem

• Definition of mechanical work by a constant force:
  W = F s \cos\theta
  • F: magnitude of force; s: displacement; \theta: angle between \vec F and \vec s.

• Work–KE Theorem (derived via Newton’s 2nd law):
  W{\text{net}} = \Delta K = \frac12 m vf^2 - \frac12 m v_0^2
  • Positive net work → object speeds up.
  • Negative net work → object slows down.

• Concept example (Question 4): Work W0 required to accelerate 0 → 50 km/h; accelerating 50 → 150 km/h demands larger work due to squared‐speed dependence (answer: 8 W0).

• Extended quantitative application (Question 5 – Deep Space 1): Small constant thrust acting over huge displacement raises probe speed; requires integrating work to obtain final kinetic energy.

• Slope–skier problem (Question 6): Combines gravitational work minus frictional work to compute final speed via \Delta K = Wg + W{\text{fric}}.

Potential Energy (PE)

• Stored energy associated with configuration or position of interacting bodies / fields.

Gravitational Potential Energy (GPE)

• Near Earth’s surface (uniform g):
  Ug = m g h (with reference level h=0 chosen arbitrarily). Often expressed as \Delta Ug = m g \Delta h.

• Depends on two factors: mass and height above reference level.

• Roller-coaster example: higher height → larger GPE; greater mass → larger GPE.

• Ranking exercise (Question 7) calls on mg\Delta h comparison for several balls.

Work Done by Gravity

• Gravity does work as an object moves vertically:
  • Displacement with gravitational force (down): \theta=0^\circ → Wg=+mgs=+mg(h0-hf)=-\Delta Ug.
  • Displacement opposite gravitational force (up): \theta=180^\circ → Wg=-mgs=-mg(h0-hf)=-\Delta Ug.

• Key takeaway: Wg = -\Delta Ug – work depends only on initial and final heights, not path.

• If a ball is thrown upward (Question 8), gravity’s work is negative (it removes kinetic energy).

• Path independence illustrated in angled‐throw question (Question 9): gravity’s work equals mg(hf-h0) regardless of arc.

• Lunar vs Earth ladder climb (Question 10): \Delta Ug larger on Earth because g{\text{Earth}} > g_{\text{Moon}}.

Elastic / Spring Potential Energy (SPE or EPE)

• Arises from stretching or compressing an ideal spring:
  U_s = \frac12 k (\Delta s)^2
  • k: spring constant (stiffness).
  • \Delta s: deformation from equilibrium (positive for stretch or compression).

• Non-linear scaling: doubling displacement quadruples energy (Question 12 → 4 × work when stretch doubled from 1 cm to 2 cm).

• Bow & arrow analogy: \Delta s equals how far archer pulls string; energy stored becomes projectile kinetic on release.

• Vertical‐spring + hanging mass (Question 11): weight stretches spring → GPE of mass decreases while SPE increases (option B).

Mechanical Energy

• Sum of kinetic and potential forms relevant to macroscopic motion:
  E{\text{mech}} = K + U (typically Ug, U_s, etc.).

• For an isolated system under conservative forces only, E_{\text{mech}} remains constant.

• When non-conservative forces (friction, thrust, etc.) act, mechanical energy is not conserved, but total energy (including thermal, chemical, etc.) still obeys conservation.

Circular vs Elliptical Orbital Work Questions

• Question 1: Moon in circular orbit with constant speed & radius → Earth’s gravitational force is perpendicular to instantaneous displacement (centripetal), so W=0 (option C, no work).

• Question 2: For elliptical orbit, speed and radius vary; force is not strictly perpendicular everywhere → Earth can do positive or negative work depending on location in orbit (option C: could be either A or B).

Summary of Conceptual & Quantitative Quick-Checks

1. Earth’s work on Moon (circular) ⇒ No work.

2. Earth’s work on Moon (elliptical) ⇒ Could be positive or negative depending on segment.

3. Tripling speed multiplies KE by 9.

4. 0 → 50 km/h needs W0; 50 → 150 km/h needs 8 W0.

5. Deep Space 1 propulsion – apply work–energy to find final speed after long thrust.

6. Skier downhill – gravitational work minus friction to compute new speed.

7. Rank GPEs via mg\Delta h.

8. Gravity does negative work on upward-thrown ball.

9. Gravity’s work path-independent; equals mg(hf-h0) even for angled trajectory.

10. Greater GPE change on Earth than Moon for same height climb.

11. Vertical spring: SPE increases, GPE decreases when mass stretches spring.

12. Work to stretch spring scales with square of stretch: 4× for double displacement.

These notes capture every definition, law, equation, conceptual nuance, and illustrative question from the transcript, synthesizing them into a study-ready outline suitable for exam preparation. Numerical factors, conservation statements, and path-independence principles are explicitly highlighted to foster deeper conceptual understanding and problem-solving proficiency.