Simple Harmonic Motion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

SHM Definition and Core Equations:

Simple Harmonic Motion is the motion where the restoring force is directly proportional to the displacement from equilibrium and always directed toward the equilibrium position: F = -kx, which gives the differential equation d²x/dt² + (k/m)x = 0. The solution is x = A cos(ωt + φ) or x = A sin(ωt + φ), where ω = √(k/m) is the angular frequency.

Essential Kinematic Quantities:

• Displacement: x = A cos(ωt + φ)
• Velocity: v = dx/dt = -Aω sin(ωt + φ); maximum v = Aω at x = 0
• Acceleration: a = dv/dt = -Aω² cos(ωt + φ) = -ω²x; maximum a = Aω² at x = ±A

Time Period Formulas for Common Systems:

System	Formula	Conditions
Spring-mass (horizontal)	T = 2π√(m/k)	Mass m, spring constant k
Simple pendulum	T = 2π√(L/g)	Small amplitude θ₀ << 1 rad, length L
Physical pendulum	T = 2π√(I/mgd)	d = distance from pivot to COM
LC circuit	T = 2π√(LC)	Pure inductance and capacitance
Liquid in U-tube	T = 2π√(h/g)	h = height of liquid column

Energy in SHM: Total mechanical energy E = ½kA² = ½mω²A². This remains constant (conserved). At displacement x: PE = ½kx² = ½mω²x², KE = ½k(A² - x²). At x = 0 (equilibrium): KE_max = ½kA², PE = 0. At x = ±A (extreme): KE = 0, PE_max = ½kA².

⚡ JEE Tip: At extremes (x = ±A): v = 0, a = ±Aω² (maximum magnitude), KE = 0. At centre (x = 0): v = ±Aω (maximum), a = 0, KE = maximum. This velocity-acceleration phase relationship is frequently tested.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Damped Simple Harmonic Motion:

Real oscillations die out because of damping forces (friction, air resistance). For a damped oscillator with damping force proportional to velocity: F_damping = -bv, the equation becomes m d²x/dt² + b dx/dt + kx = 0.

The solution is x = A₀ e^(-bt/2m) cos(ω’t + φ), where ω’ = √(ω₀² - (b/2m)²) is the damped angular frequency.

The amplitude decays exponentially: A(t) = A₀ e^(-bt/2m). The quality factor Q = ω₀ / (2β) where β = b/(2m) is the damping constant. For light damping, Q ≈ ω₀τ where τ is the time constant.

Forced Vibrations and Resonance:

When a periodic external force F = F₀ cos(ω_dt) drives a damped oscillator, the system eventually reaches steady-state oscillation at the driving frequency ω_d (not the natural frequency ω₀). The steady-state solution is x = A(ω_d) cos(ω_dt - δ) where:

A(ω_d) = F₀/m / √[(ω₀² - ω_d²)² + (bω_d/m)²]

Resonance occurs when the driving frequency ω_d ≈ ω₀, producing maximum amplitude. The resonant peak is sharper for smaller damping (higher Q). At resonance, the phase lag δ = 90°.

⚠️ Excessive damping eliminates the resonance peak — overdamped systems respond slowly but without oscillation.

⚡ JEE Tip: Spring-mass systems in series and parallel:

• Series: effective k_eq = (k₁k₂)/(k₁ + k₂), so T_series = 2π√(m/k_eq) increases (slower oscillation)
• Parallel: effective k_eq = k₁ + k₂, so T_parallel = 2π√(m/k_eq) decreases (faster oscillation) This is a common JEE problem type — attach two springs to a mass, or to the ceiling, in different arrangements.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Derivation of SHM from Energy Conservation:

For a spring-mass system, total mechanical energy E = ½mv² + ½kx² = constant. Differentiating with respect to time: m v dv/dx + kx = 0 (since v = dx/dt, dv/dt = v dv/dx). This gives m d²x/dt² + kx = 0, confirming SHM with ω = √(k/m).

Physical Pendulum — Complete Analysis:

A physical (compound) pendulum of mass M, moment of inertia I about the pivot, and distance d from pivot to centre of mass, oscillates under gravity. The restoring torque about the pivot is τ = -Mgd sinθ ≈ -Mgdθ for small angles (using small angle approximation sinθ ≈ θ). From Στ = Iα: -Mgdθ = I d²θ/dt², giving d²θ/dt² + (Mgd/I)θ = 0. Therefore T = 2π√(I/(Mgd)).

Lissajous Figures — Two Perpendicular SHMs:

When two simple harmonic motions occur perpendicular to each other, the resultant path is a Lissajous figure. The shape depends on the frequency ratio ω_x:ω_y and the phase difference δ:

• ω_x:ω_y = 1:1, δ = 0 or π → straight line (y = x or y = -x)
• ω_x:ω_y = 1:1, δ = π/2 → circle
• ω_x:ω_y = 2:1, δ = π/2 → parabola-like figure (figure of 8)
• ω_x:ω_y = m:n (integers) → closed curves (Lissajous figures)

For closed curves to form, the frequency ratio must be rational. Lissajous figures are used to compare frequencies and phase differences in AC circuits and signal analysis.

LC Oscillations — Electrical Analogue:

In an LC circuit, charge on the capacitor oscillates: q = Q₀ cos(ωt) where ω = 1/√(LC). The current is I = -I₀ sin(ωt) = dq/dt. This is mathematically identical to a spring-mass mechanical oscillator, with L analogous to mass m, 1/C analogous to spring constant k, charge q analogous to displacement x, and current I analogous to velocity v. The total energy in the LC circuit is ½(Q²/C) + ½LI² = constant, analogous to PE + KE.

⚡ JEE Advanced Pattern: JEE Advanced frequently combines SHM with rotational motion (physical pendulum), electrostatics (dipole oscillations), or optics (interference patterns). The 2023 Paper 2 Qn 38 involved a particle undergoing SHM attached to a spring, placed in a non-uniform gravitational field — students needed to find the new equilibrium position first, then redefine the coordinate system about equilibrium before identifying the SHM parameters. Common pitfalls include forgetting to shift the coordinate origin to the new equilibrium position when external forces modify the restoring force.