Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 5 | Energy in Simple Harmonic Motion

1. Why Energy Approach is Important in SHM

Energy method is the fastest and safest way to solve many SHM problems, especially in IIT/JEE.

Reduces lengthy calculations
Avoids sign confusion
Helps in maximum–minimum questions

Topper Insight:
When motion becomes confusing, switch to energy.

2. Potential Energy in SHM

In SHM, the restoring force is:

F = −kx

Potential energy is defined as:

U = ½ kx²

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

Exam Trap:
Do NOT write U = mgh for SHM problems.

3. Kinetic Energy in SHM

Velocity at displacement x is:

v = ω√(A² − x²)

Kinetic energy:

K = ½ mω² (A² − x²)

Maximum at mean position
Zero at extreme positions

At mean position:
K_max = ½ mω²A²

4. Total Mechanical Energy

Total energy in SHM is constant:

E = K + U = ½ kA² = ½ mω²A²

Independent of time
Independent of displacement

Very Important:
Energy depends on amplitude, not on position.

5. Energy Exchange During Motion

During SHM:

Potential energy converts into kinetic energy
Total energy remains constant
System oscillates due to energy exchange

At different positions:

x = 0 → K max, U = 0
x = A → U max, K = 0
x = A/√2 → K = U

IIT Favourite:
Find position where K = U.

6. Graphical View of Energy

U–x graph → upward parabola
K–x graph → downward parabola
Total energy → straight horizontal line

These graphs help in visual elimination of options.

7. Effect of Amplitude on Energy

E ∝ A²
Doubling amplitude → Energy becomes 4 times
Reducing amplitude → Energy decreases sharply

Quick Calculation Tip:
Compare energies using ratio method.

8. Common Energy-Based Mistakes

Forgetting zero potential at mean position
Using wrong expression for velocity
Assuming energy changes with time
Confusing SHM energy with circular motion energy

9. Why Energy in SHM Decides Rank

Used in multi-concept problems
Fastest method in JEE Advanced
Less algebra, more clarity

Topper Rule:
If forces confuse you, shift to energy instantly.

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