Simple harmonic motion (SHM) is the archetypal form of periodic motion. In SHM, an object oscillates about an equilibrium position, experiencing a restoring force that is directly proportional to its displacement and directed toward the equilibrium. Two classical systems that exhibit SHM are the simple pendulum (for small angles) and the mass–spring system.

1.1 Characteristics of Oscillatory Motion
1. Oscillation and Equilibrium. An oscillation refers to any repetitive back-and-forth motion about an equilibrium position (the point of zero net torque or zero net force). A complete oscillation (or cycle) is one full trip from, for example, maximum displacement on one side, through equilibrium, to maximum displacement on the other side, and back again.
2. Amplitude (A). The amplitude is the maximum displacement from equilibrium. It is always a positive quantity. In SHM, regardless of amplitude, the form of the motion remains sinusoidal, as long as the restoring force remains proportional to displacement.
3. Period (T) and Frequency (f). The period T is the time taken for one complete oscillation (units: seconds, s). The frequency f is the number of oscillations per unit time (units: hertz, Hz). They are related by f=1/T, T=1/f.
4. Angular Frequency (ω). Sometimes it is convenient to work with the angular frequency ω, defined by ω=2πf=2π/T, with units rad·s⁻¹.
5. Displacement, Velocity, and Acceleration in SHM. If x(t) is the displacement from equilibrium at time t, then in ideal SHM we write x(t)=Acos(ωt+φ), where φ is the phase constant. The velocity is v(t)=dx/dt=−Aωsin(ωt+φ), and the acceleration is a(t)=d²x/dt²=−Aω²cos(ωt+φ)=−ω²x(t).

1.2 Mass–Spring System
Consider a block of mass m attached to a horizontal, ideal (massless and frictionless) spring of spring constant k. The block can move along a frictionless surface. Let x denote the displacement from the spring’s relaxed (equilibrium) length.
1. Restoring Force. According to Hooke’s law, the spring exerts a restoring force F_spring = −k x.
2. Solution and Angular Frequency. The standard form of SHM is d²x/dt² + ω² x = 0, so by comparison, ω² = k/m. Thus, ω = √(k/m), T = 2π√(m/k), f = 1/(2π)√(k/m).
3. Energy in the Mass–Spring System. Potential Energy (Elastic). U_spring(x) = (1/2) k x². Kinetic Energy (Block). K = (1/2) m v². Total Mechanical Energy (Conserved). E_total = K + U_spring.

1.3 Simple Pendulum (Small-Angle Approximation)
A simple pendulum consists of a point mass m (the “bob”) suspended from a fixed point by a massless, inextensible string of length L. When displaced by a small angle θ (measured from vertical) and released, the bob executes (approximately) SHM.
1. Equation of Motion. Let θ(t) denote the angular displacement (in radians). The tangential force provides the restoring torque. For small angles, F_tangent ≈ −mgθ. The equation of motion results in d²θ/dt² + (g/L)θ = 0.
2. Energy in the Pendulum. The potential energy when at angle θ is U(θ) = mg(L - Lcosθ) ≈ (1/2)mgLθ², and kinetic energy is K = (1/2)m v².

1.4 General Mathematical Form of SHM
For any system undergoing SHM, one may write the displacement x(t) in one of two equivalent forms:
1. Cosine Form: x(t) = A cos(ωt + φ).
2. Sine Form: x(t) = A sin(ωt + ψ), where ψ = φ - π/2. From x(t), one can derive expressions for velocity v(t) and acceleration a(t): v(t) = dx/dt = -Aωsin(ωt + φ), a(t) = d²x/dt² = -Aω²cos(ωt + φ) = -ω²x(t).

1.5 Energy in SHM: General Considerations
Consider a one-dimensional SHM system described by x(t)=Acos(ωt+φ). Then: 1. Potential Energy (General Form). If the restoring force is F=−k_eff x, the potential energy is U(x)=(1/2) k_eff x². But ω²=k_eff/m, so one may also write U(t)=(1/2)mω² x²(t). 2. Kinetic Energy. K(t)=(1/2)m v²(t)=(1/2)m(Aωsin(ωt+φ))². 3. Total Energy (Constant). E_total=U+K=(1/2)mω² A².