15.1 Simple Harmonic Motion

Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by $f = \frac{1}{T}$ .
Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
Maximum displacement is the amplitude A. The angular frequency $ω$ , period T, and frequency f of a simple harmonic oscillator are given by $ω = \sqrt{\frac{k}{m}}$ , $T = 2 π \sqrt{\frac{m}{k}}, and f = \frac{1}{2 π} \sqrt{\frac{k}{m}}$ , where m is the mass of the system and k is the force constant.
Displacement as a function of time in SHM is given by $x (t) = A cos (\frac{2 π}{T} t + ϕ) = A cos (ω t + ϕ)$ .
The velocity is given by $v (t) = − A ω sin (ω t + ϕ) = − v_{max} sin (ω t + ϕ),$ where $v_{max} = A ω = A \sqrt{\frac{k}{m}}$ .
The acceleration is $a (t) = − A ω^{2} cos (ω t + ϕ) = − a_{max} cos (ω t + ϕ)$ , where $a_{max} = A ω^{2} = A \frac{k}{m}$ .

15.2 Energy in Simple Harmonic Motion

The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by $U = \frac{1}{2} k x^{2} .$
Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
$E_{Total} = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2} = \frac{1}{2} k A^{2} = constant.$
The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
$| v | = \sqrt{\frac{k}{m} (A^{2} - x^{2})} .$

A projection of uniform circular motion undergoes simple harmonic oscillation.
Consider a circle with a radius A, moving at a constant angular speed $ω$ . A point on the edge of the circle moves at a constant tangential speed of $v_{max} = A ω$ . The projection of the radius onto the x-axis is $x (t) = A cos (ω t + ϕ)$ , where $(ϕ)$ is the phase shift. The x-component of the tangential velocity is $v (t) = − A ω sin (ω t + ϕ)$ .

A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about $15 °$ . The period of a simple pendulum is $T = 2 π \sqrt{\frac{L}{g}}$ , where L is the length of the string and g is the acceleration due to gravity.
The period of a physical pendulum $T = 2 π \sqrt{\frac{I}{m g L}}$ can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
The period of a torsional pendulum $T = 2 π \sqrt{\frac{I}{κ}}$ can be found if the moment of inertia and torsion constant are known.

Damped harmonic oscillators have non-conservative forces that dissipate their energy.
Critical damping returns the system to equilibrium as fast as possible without overshooting.
An underdamped system will oscillate through the equilibrium position.
An overdamped system moves more slowly toward equilibrium than one that is critically damped.

A system’s natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces.
A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.