JEE/NEET Physics · Oscillations & Waves series · Part 2 of 8 · All parts →
- Total SHM energy = ½kA² (or ½mω²A²) — fixed by amplitude alone
- Energy endlessly trades: all-kinetic at centre ↔ all-potential at extremes
- Average KE = average PE = half the total
- Doubling the amplitude QUADRUPLES the energy
- Energy rides the square of amplitude — like KE rides v², spring PE rides x²
A guitar string at maximum loudness, a swing at its highest point, a trampoline at full stretch — all three have banked their energy. A moment later it’s all motion. This card puts numbers on the endless trade you first met in WEP Part 4 — now in oscillation form. Part 2 of the Oscillations & Waves series.
- The two energies, traded
- What each letter means
- The total: fixed by amplitude
- The average surprise: half and half
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Two Energies, Traded
At the extremes (±A): displacement maximum → spring/height PE maximum, motion zero. At the centre: displacement zero → PE minimum, speed maximum (v = Aω). Between: a smooth handover, every instant:
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| A | amplitude — the swing size | m |
| k | stiffness of the spring (k = mω² for SHM) | N/m |
| PE | stored energy at displacement x: ½kx² | J — max at ±A |
| KE | motion energy: ½mv² | J — max at centre |
The Total: Fixed by Amplitude
Read ½kA² closely: the square of amplitude rules the energy. Double the swing → 4× the energy; triple it → 9×. This is why loud strumming carries much more energy than gentle playing — and why a trampoline at full stretch bites harder than a gentle bob.
The Average Surprise: Half and Half
Over a full cycle, the energy spends equal time near both forms: average KE = average PE = ¼kA² — exactly half the total each. NEET loves this line: ‘in SHM, the average kinetic energy equals the average potential energy.’
Solved Examples
Direct: ½kA² = ½ × 200 × (0.05)² = 0.25 J.
Double the amplitude: ½ × 200 × 0.01 = 1.0 J — ×4, the square. ✔
Answer: 0.25 J (1.0 J if A doubles)
Energy ledger: total = ½(200)(0.1²) = 1 J. PE at 5 cm = ½(200)(0.05²) = 0.25 J → KE = 0.75 J.
½(0.5)v² = 0.75 → v = √3 ≈ 1.73 m/s.
Check: v_max = Aω = 0.1√(400) = 2 m/s — at half-displacement we’re slower, 1.73 < 2 ✔
Answer: v = √3 ≈ 1.73 m/s
The half-and-half surprise: ¼kA² : ¼kA² : ½kA² → 1 : 1 : 2.
Instantaneous values swing wildly; the averages sit perfectly level. ✔
Answer: 1 : 1 : 2
- Using x instead of A in the total. ½kx² is the PE at displacement x; the TOTAL is ½kA² — only the maximum displacement enters.
- Forgetting cm→m before squaring. 5 cm must be 0.05 m; one slip = 10,000× error.
- Assuming energy oscillates symmetrically about half. It does — that’s the 1:1:2 average rule; don’t confuse instant values with averages.
- Expecting amplitude to change the period via energy. More energy does NOT mean faster oscillation — period stays amplitude-free (Part 1).
This Physics in Your Daily Life
- Loudness of any instrument tracks energy ∝ A² — strum 41% harder for double loudness (×1.41 amplitude); ‘twice as loud’ is far more work than it sounds.
- Earthquake design: buildings store and trade SHM energy as they sway — engineers compute ½kA² equivalents to size dampers (Part 4).
- Waves at the beach: wave energy goes as height-squared — a 2× taller breaker hits with 4× the energy. Surfers and sea-walls both respect the square.
- Microwave ovens pump oscillating energy into water molecules — the molecular SHM becomes heat in your food.
- The death of loudness: as a bell rings down, energy drains ∝ A² — loudness fades fast first, then slowly: amplitude-squared, heard in every decaying note.
Practice set (answers hidden — try first)
(NEET-level) k = 100 N/m, A = 10 cm. Total energy:
(NEET-level) Amplitude doubled. Total energy becomes:
(JEE Main-level) In SHM, average KE : average PE:
(JEE Main-level) Total 2 J; at x = A/2, KE is:
(Concept) Where is PE minimum in SHM?
- 🧠 Chant: ‘energy rides the square — of amplitude, of speed, of stretch’.
- 🧠 The 1:1:2 rule: average KE : average PE : total.
- 🏠 Daily: twice-as-loud music needs ~4× the energy — amplitude-squared at the party.
- 🏠 Daily: a double-height wave breaks with 4× the punch — the square, at the beach.
- 🔁 total SHM energy = ½kA² (J)
- 🔁 energy trades: KE-max at centre, PE-max at ends
- 🔁 avg KE = avg PE = half-each of total
- total = ½kA² = ½mω²A² — amplitude-squared
- all-motion at centre (v = Aω); all-stored at extremes (a = ω²A)
- average KE = average PE = ¼kA² (1:1:2)
- doubling amplitude quadruples energy
- the trade is WEP Part 4’s see-saw in oscillation form
Quick revision
- Total SHM energy = ½kA² (or ½mω²A²) — fixed by amplitude alone
- Energy endlessly trades: all-kinetic at centre ↔ all-potential at extremes
- Average KE = average PE = half the total
- Doubling the amplitude QUADRUPLES the energy
- Energy rides the square of amplitude — like KE rides v², spring PE rides x²
- The total: fixed by amplitude
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