📚 IIT–JEE Physics Complete Learning Hub

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🔹 Core Physics Libraries

• 🔁 Simple Harmonic Motion (SHM) – Complete Library (IIT–JEE Advanced)
• 🌊 Mechanical Waves – JEE Advanced Objective Library
• 🧠 IIT–JEE Physics Complete Master Library

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🔹 Central Learning Platforms

• 🌐 Shaktimatha Learning Resource Hub
• 💬 Join Our WhatsApp Physics Learning Channel

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📌 Structured • Exam-Oriented • IIT–JEE Advanced Focused Learning

Stage 3 – Page 4
Simple Harmonic Motion (SHM) – Calculus & Graph Intelligence

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1. SHM Seen Through Calculus

SHM is best understood as a calculus-based motion, not just formulas.

x = A sin(ωt + φ)

Velocity is the time derivative of displacement:

v = dx/dt = Aω cos(ωt + φ)

Acceleration is the second derivative:

a = d²x/dt² = −Aω² sin(ωt + φ)

✔ Acceleration ∝ −displacement → SHM condition

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2. Why SHM is Called Restoring Motion

Restoring force always pulls the particle towards mean position.

F = −kx

• If x > 0 → Force is negative
• If x < 0 → Force is positive

IIT Insight: The minus sign is the soul of SHM.

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3. Graphs in SHM (Very High Weightage)

➤ Displacement–Time Graph

• Sine or cosine curve
• Slope = velocity

➤ Velocity–Time Graph

• Cosine or sine curve
• Slope = acceleration

➤ Acceleration–Time Graph

• Opposite phase to displacement

One graph can generate all others using calculus

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4. Area Under Curve Concepts (Advanced)

Area under velocity–time graph gives displacement:

∫v dt = x

Area under acceleration–time graph gives velocity:

∫a dt = v

IIT loves area-based logic questions

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5. Phase Difference from Graphs

• Displacement & velocity → π/2 phase difference
• Displacement & acceleration → π phase difference
• Velocity & acceleration → π/2 phase difference

x → v → a (phase lead by π/2 each step)

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6. Turning Points Logic (IIT Favorite)

Turning point = velocity zero.

• v = 0
• x = ±A
• a = maximum

Particle does NOT stop; it reverses direction

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7. Common Graph Traps

• Zero slope ≠ zero value
• Zero velocity ≠ zero acceleration
• Maximum displacement ≠ maximum speed

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Stage 3 – Page 4 Key Takeaways

✔ SHM is derivative-controlled motion
✔ Graphs test thinking, not memory
✔ Area concepts unlock advanced problems

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— Stage 3 | SHM Advanced Reasoning —