harmonic oscillator

Examples of harmonic oscillator in the following topics:

• Applications of Second-Order Differential Equations

  • In this atom, we will learn about the harmonic oscillator, which is one of the simplest yet most important mechanical system in physics.
  • In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, $F$, proportional to the displacement, $x$: $\vec F = -k \vec x \,$, where $k$ is a positive constant.
  • If $F$ is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
  • Driven harmonic oscillator: Driven harmonic oscillators are damped oscillators further affected by an externally applied force $F(t)$.
  • A solution of damped harmonic oscillator.

• Driven Oscillations and Resonance

  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force.
  • If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator.
  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
  • Steady state variation of amplitude with frequency and damping of a driven simple harmonic oscillator.
  • Describe a driven harmonic oscillator as a type of damped oscillator

• Energy in a Simple Harmonic Oscillator

  • The total energy in a simple harmonic oscillator is the constant sum of the potential and kinetic energies.
  • To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have.
  • Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy (KE).
  • In the case of undamped, simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates.
  • Explain why the total energy of the harmonic oscillator is constant

• The Simple Pendulum

  • A simple pendulum acts like a harmonic oscillator with a period dependent only on L and g for sufficiently small amplitudes.
  • For small displacements, a pendulum is a simple harmonic oscillator.
  • Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator.
  • For angles less than about 15º, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.
  • If θ is less than about 15º, the period T for a pendulum is nearly independent of amplitude, as with simple harmonic oscillators.

• Sinusoidal Nature of Simple Harmonic Motion

  • The solutions to the equations of motion of simple harmonic oscillators are always sinusoidal, i.e., sines and cosines.
  • Recall that the projection of uniform circular motion can be described in terms of a simple harmonic oscillator.
  • The equations discussed for the components of the total energy of simple harmonic oscillators may be combined with the sinusoidal solutions for x(t), v(t), and a(t) to model the changes in kinetic and potential energy in simple harmonic motion.
  • The others vary with constant amplitude and period, but do no describe simple harmonic motion.
  • The position of the projection of uniform circular motion performs simple harmonic motion, as this wavelike graph of x versus t indicates.

• Forced Vibrations and Resonance

  • After driving the ball at its natural frequency, the ball's oscillations increase in amplitude with each oscillation for as long as it is driven.
  • In real life, most oscillators have damping present in the system.
  • These features of driven harmonic oscillators apply to a huge variety of systems.
  • At other speeds, it is difficult to feel the bumps at all. shows a photograph of a famous example (the Tacoma Narrows Bridge) of the destructive effects of a driven harmonic oscillation.
  • The amplitude of a harmonic oscillator is a function of the frequency of the driving force.

• Damped Harmonic Motion

  • Over time, the damped harmonic oscillator's motion will be reduced to a stop.
  • The simple harmonic oscillator describes many physical systems throughout the world, but early studies of physics usually only consider ideal situations that do not involve friction.
  • We simply add a term describing the damping force to our already familiar equation describing a simple harmonic oscillator to describe the general case of damped harmonic motion.
  • Illustrating the position against time of our object moving in simple harmonic motion.
  • Describe the time evolution of the motion of the damped harmonic oscillator

• Oscillator Strengths

  • A classical harmonic oscillator driven by electromagnetic radiation has a cross-section to absorb radiation of
  • Except for the degeneracy factors for the two states, the Einstein coefficients will be the same, so we can define an oscillator strength for stimulated emission as well,
  • There are several summation rules that restrict the values of the oscillator strengths,
  • We can also separate the emission from absorption oscillator strengths

• Simple Harmonic Motion

  • It can serve as a mathematical model of a variety of motions, such as the oscillation of a spring.
  • A system that follows simple harmonic motion is known as a simple harmonic oscillator.
  • where m is the mass of the oscillating body, x is its displacement from the equilibrium position, and k is the spring constant.
  • Using Newton's Second Law, Hooke's Law, and some differential Calculus, we were able to derive the period and frequency of the mass oscillating on a spring that we encountered in the last section!
  • Relate the restoring force and the displacement during the simple harmonic motion

• Harmonic Wave Functions

  • When vibrations in the string are simple harmonic motion, waves are described by harmonic wave functions.
  • In this Atom we shall consider wave motion resulting from harmonic vibrations and discuss harmonic transverse wave in the context of a string.
  • In such condition, if we oscillate the free end in harmonic manner, then the vibrations in the string are simple harmonic motion (SHM), perpendicular to the direction of wave motion.
  • (Read our Atom on "Mathematical Representation of a Traveling Wave. ") For the case of harmonic vibration, we represent harmonic wave motion in terms of either harmonic sine or cosine function:
  • We know that time period in SHM is equal to time taken by the particle to complete one oscillation.