Work, Energy and Power

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Work, Energy and Power — Key Facts for BUET Work: W = F·d = Fd cos θ (θ = angle between F and d); unit: joule (J) Kinetic Energy: KE = ½mv²; gravitational PE: PE = mgh Work-Energy theorem: net work done = change in KE = ΔKE Conservation of mechanical energy: total energy (KE + PE) is conserved in absence of non-conservative forces ⚡ Exam tip: BUET frequently tests work-energy conservation — if only conservative forces act, KE₁ + PE₁ = KE₂ + PE₂!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Work, Energy and Power — BUET Study Guide

Work:

• W = F·d = Fd cos θ
• Only component of force along displacement does work
• Negative work when force opposes displacement (friction, gravity on upward motion)
• Unit: joule (J) = N·m

Energy types:

• Kinetic Energy (KE): KE = ½mv² (motion)
• Gravitational Potential Energy: PE = mgh (height h above reference)
• Elastic Potential Energy: PE = ½kx² (spring with stiffness k)
• Work done by variable force: W = ∫ F·dx

Work-Energy theorem: Net work done by all forces = change in KE W_net = ½mv₂² − ½mv₁²

Conservation of mechanical energy: In absence of non-conservative forces (friction, air resistance): KE₁ + PE₁ = KE₂ + PE₂

Power:

• Average power = work/time = energy/time
• Instantaneous power = F·v
• Unit: watt (W) = J/s
• 1 horsepower = 746 W

Conservative vs non-conservative forces:

• Conservative: gravity, spring force — path independent, PE definable
• Non-conservative: friction, air resistance — energy lost as heat

Spring potential energy: PE = ½kx² where k = spring constant (stiffness) x = displacement from equilibrium

Gravitational PE: PE = mgh; depends on height h above chosen zero level

Kinetic energy formula derivation: For constant force F = ma, using v² = u² + 2as: W = Fs = mas = ½m(v² − u²) = ½mv² − ½mu² = ΔKE ✓

• Key formula: W = Fd cos θ; KE = ½mv²; PE = mgh; energy conservation: KE₁ + PE₁ = KE₂ + PE₂
• Common trap: Work done by gravity going up is negative (−mgh); PE increases while KE decreases
• Exam weight: 2–3 questions per exam (8–12 marks); frequently combined with motion problems

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Work, Energy and Power — Comprehensive BUET Notes

Variable force work: W = ∫_{x₁}^{x₂} F(x) dx For spring: W = ∫₀ˣ kx dx = ½kx²

Power and velocity: P = F·v = Fv cos θ For motion along line: P = Fv (if F and v in same direction)

Elastic collision: Both momentum and kinetic energy conserved For two masses m₁, m₂ with velocities u₁, u₂: v₁ = [(m₁ − m₂)u₁ + 2m₂u₂]/(m₁ + m₂) v₂ = [(m₂ − m₁)u₂ + 2m₁u₁]/(m₁ + m₂)

Inelastic collision: Only momentum conserved; KE lost = ½μv_rel² where μ = m₁m₂/(m₁+m₂)

Vertical circular motion energy: At top of circle (radius R): minimum speed v = √(gR) Total energy at top = ½m(gR) + mg(2R) = ½mgR + 2mgR = 2.5mgR At bottom: minimum KE needed = 2mgR (to reach top with v = √(gR))

Simple pendulum: PE = mgL(1 − cos θ) KE = ½m(Lθ̇)² Total E = mgL(1 − cos θ) + ½mL²θ̇² = constant

Mass-spring system: x = A cos(ωt + φ) where ω = √(k/m) KE = ½mω²A² sin²(ωt + φ) PE = ½kA² cos²(ωt + φ) Total = ½kA² (constant)

Reduced mass: For two-body system: μ = m₁m₂/(m₁ + m₂) Replaces two-body problem with single body of mass μ

Conservative force and potential: F = −dU/dx (one dimension) In 3D: F = −∇U For gravity: U = mgh; F = −dU/dh = −mg (downward) For spring: U = ½kx²; F = −dU/dx = −kx (restoring)

Non-conservative forces: Work done by friction = −f_k × distance This energy becomes heat (not recoverable as mechanical energy)

Gravitational potential: For large distances (not near Earth’s surface): U = −GMm/r (negative, since zero at infinity) For Earth’s surface: g = GM/R², so U = mgR at surface

Orbital energy: Kinetic: KE = ½mv² = GMm/(2r) Potential: PE = −GMm/r Total: E = KE + PE = −GMm/(2r) (negative, bound orbit) Escape velocity: v_esc = √(2GM/R) = √(2gR)

Time-varying force and energy: For any force: W = ΔKE (work-energy theorem, always valid) Only for conservative forces: W = −ΔPE

Power and efficiency: Efficiency η = useful output/total input For machine: η = (work output)/(work input) × 100%

Circular motion and energy: For satellite in circular orbit: v = √(GM/r) KE = GMm/(2r); PE = −GMm/r; E = −GMm/(2r)

Energy method for velocity: For conservative system: ½mv² + mgh + ½kx² = constant Use to find speed at any position without solving differential equation

• Remember: W = ∫ F·dx = ΔKE; for conservative forces: KE + PE = constant; spring PE = ½kx²; escape velocity v_esc = √(2gR)
• Previous years: “Block slides down from height h. Find speed at bottom” [2023 BUET]; “Spring with k=100 N/m compressed 0.1m. Find max speed of 2kg block” [2024 BUET]; “Find power needed to lift 100kg at 2m/s” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Physics

• Mechanics (Laws of Motion, Work-Energy, Rotational) — very high weight
• Electrostatics and Current Electricity — very high weight
• Modern Physics (Photoelectric effect, Atoms) — high weight
• Heat and Thermodynamics — medium-high weight
• Optics (Reflection, Refraction) — medium weight

📝 Previous Year Question Patterns

• Work, Energy, Power: 2–3 questions per exam, 8–12 marks
• Common patterns: energy conservation, spring PE, power calculations, collision KE
• Weight: very high — frequently combines with motion and circular motion

💡 Pro Tips

• Work-energy theorem is always valid, even with friction: W_net = ΔKE
• Only conservative forces allow mechanical energy conservation
• For springs: PE = ½kx² regardless of whether compressing or extending
• Gravitational PE = mgh only valid when g is constant; for large heights, use U = −GMm/r
• When solving energy problems, always choose reference level for PE clearly

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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