Solution: x = A sin(ωt + φ)   or   x = A cos(ωt + φ)

A = amplitude, ω = angular frequency, φ = initial phase.

Time period: T = 2π/ω     Frequency: f = 1/T = ω/2π

v = dx/dt = Aω cos(ωt + φ) = ± ω√(A² – x²)

a = d²x/dt² = –Aω² sin(ωt + φ) = –ω²x

Maximum velocity: v_max = Aω (at mean position)

Maximum acceleration: a_max = Aω² (at extreme positions)

KE = (1/2)mv² = (1/2)mω²(A² – x²)

PE = (1/2)kx² = (1/2)mω²x²

For a spring of constant k: ω = √(k/m), T = 2π√(m/k)

Series combination: 1/k_eff = 1/k₁ + 1/k₂

Parallel combination: k_eff = k₁ + k₂

Restoring force: F = –mg sinθ ≈ –mgθ (for small θ)

Effective g due to acceleration: T = 2π√(L/g_eff)

💡 NEET TIPS & SHORTCUTS

• In vertical spring, equilibrium position shifts but ω = √(k/m) remains same.
• For a spring of mass m_s, effective mass = m + m_s/3 in T formula.
• In compound pendulum, T = 2π√(I/mgd).
• For a body in SHM, average KE = average PE = E/2.

⚠️ COMMON MISTAKES

• Using ω = √(g/L) for spring instead of √(k/m).
• Forgetting to convert angle to radians in pendulum formulas.
• Assuming SHM is only sinusoidal – phase difference matters in superposition.
• Confusing angular frequency ω with angular velocity.

SHM equation: x = A sin(ωt + φ)

ω = 2π/T = 2πf

v_max = Aω, a_max = Aω²

Spring T: T = 2π√(m/k)

Pendulum T: T = 2π√(L/g)

Total energy: E = ½kA² = ½mω²A²