• Periodic motion back and forth over same path
• Restoring force proportional to displacement from equilibrium
• $F_{restoring} = -kx$ (Hooke's law type force)
• Acceleration proportional to displacement from equilibrium
• Examples: mass-spring system, simple pendulum (small angles)

• Force directed toward equilibrium position
• Proportional to displacement: $F = -kx$
• Causes oscillation about equilibrium
• Example: spring force, gravitational component for pendulum

• Position: $x = A\cos(\omega t + \phi)$
• Velocity: $v = -A\omega\sin(\omega t + \phi)$
• Acceleration: $a = -A\omega^2\cos(\omega t + \phi)$
• All vary sinusoidally with same frequency
• Phase constant $\phi$ depends on initial conditions

$T = 2\pi\sqrt{\frac{m}{k}}$

• Period depends on mass and spring constant only
• Independent of amplitude (for ideal springs)
• Independent of gravitational acceleration
• Frequency: $f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

$T = 2\pi\sqrt{\frac{L}{g}}$ (small angle approximation)

• Depends on length and gravitational acceleration
• Independent of mass and amplitude (small angles < 15 degrees )
• Period increases with length
• Larger g -> shorter period

• Mass: No effect on period (both spring and pendulum)
• Amplitude: No effect on period (small oscillations)
• Large amplitudes break approximations
• Period depends on system parameters, not initial conditions

• Length: Longer pendulum -> longer period
• Gravity: Larger g -> shorter period
• Used in pendulum clocks and gravimeters
• Depends on $L$ and $g$ only (for small angles)

• Total mechanical energy: $E = K + U = \frac{1}{2}kA^2$ (constant)
• Maximum kinetic energy at equilibrium: $K_{max} = E$
• Maximum potential energy at endpoints: $U_{max} = E$
• Energy continuously converts between kinetic and potential
• No energy loss in ideal SHM (no friction/damping)

• Both vary sinusoidally with time
• 180 degrees out of phase: when K maximum, U minimum (zero), and vice versa
• Sum is constant (conservation of energy)
• Useful for visualizing energy exchange

• Maximum velocity: At equilibrium position ($x = 0$)
• $v_{max} = A\omega$
• Maximum acceleration: At endpoints ($x = \pm A$)
• $a_{max} = A\omega^2$
• Used to find amplitude or frequency from velocity/acceleration