Energy in SHM — KE, PE, Total Energy and Damping

• Derive expressions for kinetic energy (KE) and potential energy (PE) as functions of displacement in SHM
• Show that total mechanical energy E = ½mω²A² is constant throughout the motion
• Sketch and interpret graphs of KE and PE versus displacement and versus time
• Explain what happens to KE and PE at mean position and extreme positions
• Describe damped oscillations and the effect of damping on amplitude and energy

Level 1 — Foundations

Kinetic energy in SHM At displacement x, velocity v = ω√(A² − x²), so: $KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2 - x^2)$

• At mean position (x = 0): KE = ½mω²A² (maximum)
• At extreme positions (x = ±A): KE = 0

Potential energy in SHM The restoring force is F = −kx = −mω²x. PE stored: $PE = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2$

• At mean position (x = 0): PE = 0 (minimum)
• At extreme positions (x = ±A): PE = ½mω²A² (maximum)

Total mechanical energy $E = KE + PE = \frac{1}{2}m\omega^2(A^2 - x^2) + \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}m\omega^2 A^2$ Total energy is constant — independent of displacement and time.

Key result: $\boxed{E = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2}$

Energy position graphs

• PE vs x: parabola (U-shaped), minimum at x = 0, maximum at x = ±A
• KE vs x: inverted parabola, maximum at x = 0, zero at x = ±A
• Total E vs x: horizontal straight line (constant)
• KE = PE when x = ±A/√2

Energy time graphs Since x = A sin(ωt):

• PE = ½mω²A² sin²(ωt) — oscillates between 0 and ½mω²A² with period T/2
• KE = ½mω²A² cos²(ωt) — oscillates between 0 and ½mω²A² with period T/2
• Both KE and PE oscillate at twice the frequency of the displacement

Level 2 — JEE / NEET depth

Derivation from first principles Work done by restoring force as particle moves from x₀ to x: $W = -\int_{x_0}^{x} (-m\omega^2 x)\, dx = \frac{1}{2}m\omega^2(x_0^2 - x^2)$ This equals the change in KE (work-energy theorem). Taking x₀ = A (starting from rest at extreme): $KE = \frac{1}{2}m\omega^2(A^2 - x^2) \quad \checkmark$

E ∝ A² — critical JEE result If amplitude doubles: E increases by 4×. If angular frequency doubles (at fixed A): E increases by 4×.

Position where KE = PE $\frac{1}{2}m\omega^2(A^2 - x^2) = \frac{1}{2}m\omega^2 x^2$ $A^2 - x^2 = x^2 \implies x = \pm\frac{A}{\sqrt{2}} \approx \pm 0.707A$

Damped oscillations In real systems, friction/air resistance removes energy:

• Amplitude decreases exponentially: A(t) = A₀ e^{−bt/2m} where b = damping coefficient
• Total energy decreases: E(t) = E₀ e^{−bt/m}
• Time period increases slightly with damping
• Three cases:
  • Underdamped (b < 2√(km)): oscillates with decreasing amplitude
  • Critically damped (b = 2√(km)): returns to equilibrium fastest without oscillating
  • Overdamped (b > 2√(km)): returns to equilibrium slowly without oscillating

JEE traps

• PE in SHM is ½kx², NOT ½kA² (that is the total energy, not PE at general x)
• KE is maximum at mean position, NOT at extreme — opposite of what many students assume
• Energy oscillates at 2ω (double the SHM frequency) — a common MCQ trap
• Damping reduces amplitude but does NOT immediately change the time period formula for light damping

Energy at a given displacement

A particle of mass 0.1 kg executes SHM with amplitude 10 cm and angular frequency 4 rad/s.
Find: (a) total energy, (b) KE and PE at x = 6 cm.

(a) E = ½mω²A² = ½ × 0.1 × 16 × (0.1)² = ½ × 0.1 × 16 × 0.01
      = 0.008 J = 8 mJ

(b) At x = 6 cm = 0.06 m:
    PE = ½mω²x² = ½ × 0.1 × 16 × (0.06)² = 0.5 × 0.1 × 16 × 0.0036 = 0.00288 J ≈ 2.88 mJ
    KE = E − PE = 8 − 2.88 = 5.12 mJ

    Check: KE = ½mω²(A²−x²) = ½ × 0.1 × 16 × (0.01 − 0.0036) = 0.8 × 0.0064 = 0.00512 J ✓

Finding x where KE = PE

In the above example, find displacement where KE = PE.

KE = PE ⟹ x = A/√2 = 10/√2 = 5√2 ≈ 7.07 cm

Verify: PE = ½ × 0.1 × 16 × (0.0707)² = 0.8 × 0.005 = 0.004 J = 4 mJ
        KE = 8 − 4 = 4 mJ ✓ (KE = PE = E/2)

• Q1: Write the expression for total energy in SHM and show it is constant.
• Q2: A spring (k = 200 N/m) has a mass of 0.5 kg attached. If amplitude is 4 cm, find total energy.
• Q3: At what fraction of the amplitude is KE equal to PE?
• Q4: In damped oscillations, what happens to amplitude and total energy over time?
• Stretch: Show that the time-averaged values of KE and PE in SHM are each equal to E/2.

NCERT Chapter 14 link: Section 14.5 "Energy in Simple Harmonic Motion" with derivations of KE, PE, and total energy. Figure 14.6 shows KE/PE vs displacement graphs. Section 14.8 covers damped oscillations (qualitative). Exercises 14.14–14.18 are directly on energy.

Exam connections: JEE Main: energy at a specific displacement (very common), finding x where KE = PE, E ∝ A² problems. JEE Advanced: deriving average KE and PE, energy in damped systems, comparing SHM and circular motion energy analogies. NEET: more qualitative — graph interpretation, KE at mean/extreme positions.

Study strategy: Start with the three key positions — mean (x = 0), extreme (x = ±A), and x = A/√2 — and memorise KE and PE at each. Then use E = KE + PE for any other x. Always double-check: KE + PE must equal ½mω²A² at every point.

• Use the platform-native energy simulation: watch KE and PE bar charts change as a spring-mass system oscillates; confirm they always sum to the same total.
• Home activity: stretch a rubber band to different amplitudes and feel the difference in stored energy — relate this to PE = ½kA².
• Voice or text reflection with AI Mentor: explain why a clock pendulum slows down over time without a battery — connect to damping and energy loss.

• "Explain, using energy conservation, why a pendulum's speed is maximum at the lowest point."
• "What is one common mistake students make about KE and PE in SHM, and how would you explain the correct idea to a friend?"
• Stretch: "How does the concept of damping apply to car shock absorbers — which damping type is most desirable and why?"

• Complete the core practice + one extension activity (graph sketch, calculation table, or energy diagram) for base XP + Energy in SHM badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project: attach a spring to a mass on a frictionless table (or simulate with a rubber band), measure oscillation amplitude after each swing, and plot the decay to visualise damping.
• Direct link to Future Skill track: Green Tech (energy harvesting from mechanical vibrations uses SHM energy analysis), Engineering & Robotics (damping design is critical for bridge vibration control and earthquake engineering).
• Coding extension: simulate damped SHM in Python — plot x(t), KE(t), PE(t), and E(t) on the same figure; vary the damping coefficient b and observe changes.

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: A hand-drawn or digital KE/PE vs x graph with annotations explaining each feature, plus one sentence on a real device where energy-SHM analysis matters.

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.