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Simple Harmonic Motion (SHM) – One Page Formula & Mind Map

IIT-JEE | JEE Advanced | Final Revision Capsule

1️⃣ Core Concept (Mind Anchor)

SHM Condition:
Restoring force ∝ displacement and acts towards mean position.

Mathematically: F = −kx → a = −ω²x

2️⃣ Basic Equations of SHM

Quantity	Formula
Displacement	x = A sin(ωt + φ)
Velocity	v = ω√(A² − x²)
Acceleration	a = −ω²x
Maximum Velocity	v_max = ωA
Maximum Acceleration	a_max = ω²A

3️⃣ Time Period (Most Tested)

System	Time Period (T)
Spring–Mass	T = 2π√(m/k)
Simple Pendulum	T = 2π√(ℓ/g)
Physical Pendulum	T = 2π√(I/mgd)

4️⃣ Energy Relations

Kinetic Energy: K = ½ mω²(A² − x²)

Potential Energy: U = ½ mω²x²

Total Energy (Constant): E = ½ mω²A²

5️⃣ Phase Relations (IIT Favorite)

Velocity leads displacement by π/2
Acceleration leads velocity by π/2
Acceleration is always opposite to displacement

6️⃣ High-Yield IIT Shortcuts

At mean position → v = max, a = 0
At extreme position → v = 0, a = max
Time from mean to extreme = T/4
Time between two opposite points = T/2
SHM graph area ≠ distance (don’t confuse!)

7️⃣ SHM Mind Map (Textual)

SHM → Definition → Equation (a = −ω²x) → Motion (x, v, a) → Time Period → Energy Exchange → Phase Difference → Graphs (x-t, v-t, a-t) → Pendulums → PYQs & Traps

8️⃣ Exam-Day Checklist

Check equilibrium position carefully
Never assume SHM without restoring force
Write ω first → reduces mistakes
Use energy method when force is messy
Watch phase signs (±)

✅ One-Page SHM Formula & Mind Map Completed