Rucete ✏ AP Physics C In a Nutshell
8. Simple Harmonic Motion
This chapter introduces simple harmonic motion (SHM), including the conditions under which SHM occurs, the mathematical description of SHM, key properties such as amplitude, frequency, and period, and energy conservation methods for solving SHM problems. It also explores connections between SHM and uniform circular motion (UCM).
Definition of Simple Harmonic Motion
• SHM is oscillatory motion governed by a linear restoring force or torque.
• Displacement from equilibrium is described by sine or cosine functions.
• SHM is a special case of periodic motion where the restoring force is proportional to displacement.
Mathematical Description of Mass-Spring SHM
• Hooke’s Law for springs: F = –kx
• Differential equation of motion: m(d²x/dt²) = –kx
• General solution: x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + ϕ)
• Parameters A (amplitude) and φ (phase constant) are determined by initial conditions.
• Angular frequency: ω = √(k/m)
Restoring Forces and SHM Conditions
• SHM arises when forces act to restore an object to equilibrium and are proportional to displacement.
• Small-amplitude oscillations around stable equilibrium points can be approximated linearly.
• Restoring force: F ≈ –keffective·x
• Rotational analog: τ ≈ –keffective·θ
Energy in SHM
• Total mechanical energy (E) is conserved if no nonconservative forces act:
E = ½kx² + ½mv²
• Energy oscillates between kinetic and potential forms during motion.
Frequency, Period, and Angular Frequency
• Linear frequency (f) = number of cycles per second (Hz).
• Period (T) = time for one complete cycle: T = 1/f
• Angular frequency (ω) related to f: ω = 2πf
Examples of SHM
Simple Pendulum
• Small-angle approximation (θ small): restoring force ≈ –mgθ.
• Period: T = 2π√(L/g)
• Period is independent of mass and amplitude (for small θ).
Vertical Spring Systems
• Equilibrium position shifted by mg = kΔy.
• Oscillations occur about this new equilibrium point.
• Frequency determined by mass m and spring constant k, just like horizontal springs.
Physical Pendulum
• Extended body swinging about a pivot point.
• Period: T = 2π√(I/mgD)
• I = rotational inertia about pivot, D = distance from pivot to center of mass.
Energy Conservation Applications in SHM
• At maximum displacement (x = A): all energy is potential (E = ½kA²).
• At equilibrium position (x = 0): all energy is kinetic (E = ½mv²).
• Use conservation of energy to find speeds or displacements at various points during oscillation.
Connection to Uniform Circular Motion (UCM)
• SHM can be viewed as projection of UCM onto one axis.
• Circular motion with constant speed but varying projected position and velocity.
• Mathematical analogy simplifies understanding SHM behavior (x(t) = r cos(ωt)).
In a Nutshell
Simple harmonic motion describes systems where the restoring force is proportional to displacement. The mathematics of SHM applies to springs, pendulums, and oscillating bodies, and connects naturally to uniform circular motion. Energy conservation methods offer powerful tools for analyzing oscillatory motion in both mechanical and rotational systems.