Learning Objectives
- Understand periodic, oscillatory, and simple harmonic motion
- Derive equations for displacement, velocity, and acceleration in SHM
- Study energy in SHM and its oscillation between KE and PE
- Learn about the simple pendulum and spring-mass system
- Understand damped, forced oscillations, and resonance
Key Concepts
Simple Harmonic Motion (SHM)
A periodic motion where the restoring force is directly proportional to displacement from the mean position and is directed towards it.
Condition: F = -kx (Hooke's law type restoring force)
Differential equation: d²x/dt² + ω²x = 0
Equations of SHM
Displacement: x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ)
Velocity: v(t) = Aω cos(ωt + φ) = ω√(A² - x²)
Acceleration: a(t) = -Aω² sin(ωt + φ) = -ω²x
Maximum velocity: v_max = Aω (at mean position). Maximum acceleration: a_max = Aω² (at extreme position).
where A = amplitude, ω = angular frequency, φ = initial phase, (ωt + φ) = phase.
Time period: T = 2π/ω. Frequency: f = 1/T = ω/2π.
Energy in SHM
Kinetic Energy: KE = ½mω²(A² - x²) = ½mω²A² cos²(ωt + φ)
Potential Energy: PE = ½mω²x² = ½mω²A² sin²(ωt + φ)
Total Energy: E = KE + PE = ½mω²A² = ½kA² = constant
KE and PE oscillate with frequency 2ω (twice the SHM frequency). Average KE = Average PE = E/2.
At mean position: KE = max, PE = 0. At extremes: KE = 0, PE = max.
Spring-Mass System
Time period: T = 2π√(m/k), ω = √(k/m)
Independent of amplitude and acceleration due to gravity.
Springs in series: 1/k_eff = 1/k₁ + 1/k₂
Springs in parallel: k_eff = k₁ + k₂
Simple Pendulum
Time period: T = 2π√(L/g) (for small angular amplitude θ < 15°)
Independent of mass and amplitude (for small oscillations).
Restoring force: F = -mg sin θ ≈ -mgθ (for small θ)
Effective length L = distance from pivot to centre of mass of bob.
In an accelerating lift: T = 2π√(L/g_eff), where g_eff = g + a (upward) or g - a (downward).
Damped Oscillations
Oscillations whose amplitude decreases over time due to resistive forces (like friction, air resistance).
Equation: x(t) = Ae^(-bt/2m) sin(ω't + φ), where b is the damping constant.
Damped frequency: ω' = √(ω₀² - (b/2m)²) < ω₀ (natural frequency).
Energy: E(t) = ½kA²e^(-bt/m) (exponential decay).
Forced Oscillations and Resonance
When an external periodic force drives the oscillation: F_ext = F₀ cos ωdt
Resonance: When driving frequency ωd equals natural frequency ω₀, amplitude is maximum. This is resonance.
Amplitude at resonance: A_max = F₀/(bω₀) (limited by damping).
Applications: tuning a radio, musical instruments, swinging, MRI.
Summary
SHM is characterized by a restoring force proportional to displacement. The displacement, velocity, and acceleration are sinusoidal functions. Total energy in SHM is constant and proportional to A². The simple pendulum (T = 2π√(L/g)) and spring-mass system (T = 2π√(m/k)) are classic examples. Damped oscillations lose energy over time. Forced oscillations at resonance produce maximum amplitude.
Important Terms
- Amplitude: Maximum displacement from mean position
- Phase: Argument of the sinusoidal function, determines position at any time
- Angular Frequency: ω = 2π/T = 2πf
- Resonance: Maximum amplitude when driving frequency equals natural frequency
- Damping: Decrease in amplitude due to dissipative forces
Quick Revision
- SHM: x = A sin(ωt + φ), v = Aω cos(ωt + φ), a = -ω²x
- v_max = Aω (at x=0); a_max = Aω² (at x=±A)
- E = ½kA² = ½mω²A² = constant
- Spring: T = 2π√(m/k); Pendulum: T = 2π√(L/g)
- At mean position: KE = max, PE = 0
- Resonance: ωd = ω₀ → maximum amplitude