Stage–2 | Page–2
Energy Method in Simple Harmonic Motion

When force approach becomes messy, energy saves you


Why IIT Loves Energy Method

  • No direction confusion
  • No sign mistakes
  • Works even when force is complex
IIT Tip:
If you see turning points, maximum speed, or amplitude → think energy.

Total Mechanical Energy in SHM

In ideal SHM (no friction):

Total Energy (E) = Constant

Energy has two parts:

  • Kinetic Energy (KE)
  • Potential Energy (PE)

Potential Energy in SHM

Restoring force:

F = −kx

Potential energy function:

PE = ½ kx²

Important: Zero PE is taken at equilibrium.


Kinetic Energy in SHM

KE = ½ mv²

Velocity is maximum at equilibrium, zero at extreme positions.


Total Energy Expression

At extreme position (x = A):

E = ½ kA²

This is the total energy of SHM.


Energy Distribution at Any Position x

KE = ½ k (A² − x²)
PE = ½ k x²

Sum remains constant.


Velocity–Position Relation (Very Important)

v = ω √(A² − x²)

This formula is frequently used in JEE Advanced.


Special Energy Cases (Must Remember)

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

IIT Traps Using Energy

  • Energy is scalar – direction not needed
  • Do not confuse displacement with distance
  • Amplitude may change if energy changes

Self-Check (IIT Mindset)

  • Can I find speed without time?
  • Can I find position without phase?
  • Can I locate turning points?

What Comes in Stage–2 Page–3?

  • Phase, phase constant & phase difference
  • Time-based analysis
  • Multiple SHM comparison

Stage–2 Page–2 Completed ✔ You are now dangerous to tough SHM problems 😄

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