Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 8 | Energy in SHM, KE–PE Variation & Graphs

1. Energy Concept in SHM

In Simple Harmonic Motion, energy continuously converts between Kinetic Energy (KE) and Potential Energy (PE).

However, the Total Mechanical Energy remains constant (when no damping is present).

2. Potential Energy in SHM

Potential energy of a particle executing SHM is:

PE = ½ k x²

k → force constant
x → displacement from mean position

PE is:

Zero at mean position (x = 0)
Maximum at extreme positions (x = ±A)

3. Kinetic Energy in SHM

Kinetic energy is given by:

KE = ½ m ω² (A² − x²)

KE is:

Maximum at mean position
Zero at extreme positions

Golden Result:
Maximum KE = Total Energy

4. Total Energy of SHM

Total mechanical energy is constant and given by:

E = ½ k A² = ½ m ω² A²

Independent of time
Independent of position

5. Energy Distribution at Key Positions

Position	KE	PE
Mean (x = 0)	Maximum	Zero
Extreme (x = ±A)	Zero	Maximum
x = A/√2	Equal to PE	Equal to KE

6. KE–PE vs Displacement Graphs

PE vs x → Parabolic curve opening upward
KE vs x → Inverted parabola
Total Energy → Horizontal straight line

Graph Insight:
At any x, KE + PE = constant.

7. KE–PE vs Time Graphs

KE varies as sin²(ωt)
PE varies as cos²(ωt)
Phase difference between KE & PE = π/2

This phase difference is frequently tested in IIT/JEE.

8. Velocity–Displacement Relation

Velocity of particle in SHM:

v = ω √(A² − x²)

Maximum velocity at mean position
Zero velocity at extremes

9. Common IIT/JEE Traps

Assuming KE = PE at mean position ❌
Forgetting squared terms in energy
Wrong graph orientation
Mixing time and displacement graphs

Exam Tip:
Energy questions look simple but eliminate careless students.

10. Why This Page Is Extremely Important

Direct formula-based questions
Graph interpretation questions
Conceptual + numerical mix
High accuracy, low time

Stage 1 – Page 8 Completed Successfully ✅

📚 IIT–JEE Physics Complete Library

Concept Mastery • PYQs • Strategy • Revision
One-stop structured learning hub

🔹 Simple Harmonic Motion (SHM) — Core Series

Coverage: Concepts → PYQs → Advanced Thinking → Exam Readiness

🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)

These pages are placed separately for continuity & reference.

30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56

🔹 Supporting & Mega Libraries

✔ Library logically bifurcated • ✔ All links preserved • ✔ Student-first design

MIND GROW Magazine