Complete SHM in 15 Minutes
One-page summary + flashcards + formula dump + memory tricks. Use this the night before your exam. Everything that matters, nothing that doesn't.
🧠 Flashcard Revision Mode
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📋 Formula Dump (All-in-One)
v(t) = −Aω sin(ωt + φ)
a(t) = −Aω² cos(ωt + φ)
v = ω√(A²−x²)
v_max = Aω
a_max = Aω²
f = 1/T = ω/2π
ω = 2πf = 2π/T
T(spring) = 2π√(m/k)
T(pendulum) = 2π√(L/g)
ω(spring) = √(k/m)
ω(pendulum) = √(g/L)
KE = ½mω²(A²−x²)
PE = ½mω²x² = ½kx²
KE = PE at x = ±A/√2
E ∝ A² ∝ ω²
Parallel: k_p=k₁+k₂
Cut to n parts: k_new=nk
T_new(cut to n)=T₀/√n
a leads v by: π/2
a vs x: π (opposite)
KE vs PE (time): π/2
KE,PE period = T/2
g_eff(elev up)=g+a
g_eff(horiz)=√(g²+a²)
A_R=√(A₁²+A₂²+2A₁A₂cosδ)
ω²=d²U/dx²|_eq / m
📘 One-Page Concept Summary
What is SHM?
Motion where restoring force ∝ displacement and is directed toward equilibrium. F = −kx.
Key equation: d²x/dt² + ω²x = 0 → general solution: x = A cos(ωt + φ)
3 Standard Systems
- Horizontal spring-mass: T = 2π√(m/k)
- Vertical spring-mass: Same T! (gravity shifts equilibrium only)
- Simple pendulum: T = 2π√(L/g) — small angle only
Energy Distribution
- At equilibrium (x=0): KE = max = E, PE = 0
- At extreme (x=±A): PE = max = E, KE = 0
- At x=±A/√2: KE = PE = E/2
- Total E = ½kA² = constant
Phase Relationships
5 Key Graphs
- x-t: Cosine/Sine wave
- v-t: Sine wave, 90° ahead of x-t
- a-t: Cosine wave, inverted (π behind x)
- a-x: Straight line, slope = −ω²
- v-x: Ellipse with semi-axes A and Aω
- KE-t, PE-t: Period = T/2 of SHM
SHM Identification Test
Does d²x/dt² = −(constant)×x? If yes → SHM. Angular frequency ω = √(that constant).
🧠 Memory Tricks & Mnemonics
🔢 The π/2 Staircase
Remember phase differences as a staircase:
↓ +π/2 phase
v (step 1)
↓ +π/2 phase
a (step 2) = x + π
Each derivative adds π/2 phase. After two derivatives (x → a), you've added π.
🎵 "VAX" Rule for Extremes
At Velocity = 0 → you're at extreme (x = ±A).
At Acceleration = 0 → you're at equilibrium (x = 0).
V = 0 → At eXtreme (A)
A = 0 → At eQuilibrium (0)
⚡ "SOFT" for Spring Constant
When a spring is made Shorter, k gets Only Faster (larger k, smaller T).
Longer spring → softer → larger T → slower oscillation.
Longer spring → Smaller k
k × length = constant (kL = const)
🌙 "Moon Makes Pendulum Mourn"
On Moon (smaller g) → T = 2π√(L/g) increases → pendulum swings slower → "mourning, slow pace".
Smaller g → Larger T → Slower pendulum
T_Moon/T_Earth = √(g_E/g_M) = √6 ≈ 2.45
⚡ "Equals at Root Two"
KE = PE when x = A/√2 ≈ 0.707A
Each = ½ × Total Energy
Remember: "root two divides equally"
🔄 "SEQA": Sequence of SHM States
Starting from equilibrium (+ve direction): Start → Extreme+ → Qequilibrium → Extreme− → Again start
x=0,v=−max → x=−A,v=0 →
x=0,v=max (repeat)
⚡ Rapid Fire Facts (Read Aloud Before Exam)
T does NOT depend on amplitude for ideal SHM
T does NOT depend on mass for pendulum
Vertical spring-mass: T = 2π√(m/k), NOT affected by g
Spring cut to n equal parts: each part has k_new = nk
ω = 2πf (NOT ω = f. The most common unit error.)
Phase of acceleration vs displacement = π (opposite)
Energy ∝ A². Double amplitude → 4× energy.
KE and PE oscillate at 2f (twice SHM frequency)
Mass added at extreme: amplitude stays same, T changes
Constant force on SHM: equilibrium shifts, T unchanged
Energy graph T = half of displacement graph T
Free fall pendulum: g_eff = 0, T = ∞, no oscillation
📊 Your Preparation Progress
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