Rucete ✏ AP Physics 1 In a Nutshell
8. Oscillatory Motion
This chapter covers simple harmonic motion (SHM) including mass-spring systems, pendulums, and the dynamics of oscillations. It explains how forces and energy influence periodic motion and introduces key equations for analyzing amplitude, frequency, velocity, and acceleration.
Simple Harmonic Motion: A Mass on a Spring
• Hooke’s law: F = –kx — the restoring force is proportional and opposite to displacement.
• SHM occurs when acceleration is proportional to and opposite the displacement: a = –(k/m)x
• Acceleration is zero at equilibrium (x = 0); maximum at maximum displacement (x = ±A).
• Displacement vs. time graph is a cosine function if motion begins at maximum displacement.
• Velocity is zero at endpoints (x = ±A), maximum at equilibrium.
• Energy: Total E = (1/2)kA² = constant
Potential energy (spring) = (1/2)kx²
Kinetic energy = (1/2)mv²
Period and Frequency of a Mass-Spring System
• Period: T = 2π√(m/k)
• Frequency: f = 1/T
• Angular frequency: ω = 2πf = √(k/m)
• Independent of amplitude and gravity.
Simple Harmonic Motion: A Simple Pendulum
• A pendulum swings in SHM if the angle is small (θ < 15°).
• Restoring force: F = –mg·sin(θ) ≈ –mgθ (small-angle approximation)
• Period: T = 2π√(L/g)
• Depends only on length and gravity, not mass or amplitude (for small angles).
• Like a spring-mass system, energy oscillates between kinetic and potential:
– PE is max at endpoints, zero at lowest point.
– KE is max at lowest point, zero at endpoints.
Dynamics of SHM
• Acceleration: a = –ω²x = –(k/m)x for spring system
• Velocity: v = ±ω√(A² – x²)
• Displacement: x(t) = A·cos(ωt + φ), where φ is the phase constant (depends on initial conditions)
• Energy conservation holds: Total energy is constant, exchanges between KE and PE.
• Graphical representations show sinusoidal relationships for x, v, and a.
In a Nutshell
Oscillatory motion occurs when a system moves back and forth about equilibrium. In simple harmonic motion, restoring forces are linear, leading to sinusoidal motion. Springs and pendulums exhibit SHM under specific conditions, governed by well-defined equations. Energy constantly transforms between potential and kinetic while total energy remains conserved.