simple harmonic motion

(noun)

(SHM) — Oscillating motion (as of a pendulum) in which the acceleration of the oscillator has an equal magnitude but opposite direction to the displacement of it from the equilibrium position.

Examples of simple harmonic motion in the following topics:

  • Simple Harmonic Motion

    • Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.
    • In addition, other phenomena can be approximated by simple harmonic motion, such as the motion of a simple pendulum, or molecular vibration.
    • A system that follows simple harmonic motion is known as a simple harmonic oscillator.
    • A brief introduction to simple harmonic motion for calculus-based physics students.
    • Relate the restoring force and the displacement during the simple harmonic motion

  • Simple Harmonic Motion and Uniform Circular Motion

    • Simple harmonic motion is produced by the projection of uniform circular motion onto one of the axes in the x-y plane.
    • There is an easy way to produce simple harmonic motion by using uniform circular motion.
    • The shadow undergoes simple harmonic motion .
    • Its projection on the x-axis undergoes simple harmonic motion.
    • Describe relationship between the simple harmonic motion and uniform circular motion

  • 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.

  • 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.
    • We know that a traveling wave function representing motion in x-direction has the form:
    • (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:

  • Energy in a Simple Harmonic Oscillator

    • The total energy in a simple harmonic oscillator is the constant sum of the potential and kinetic energies.
    • 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.
    • This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.
    • If we start our simple harmonic motion with zero velocity and maximum displacement (x=X), then the total energy is:
    • The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface.

  • 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.
    • Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.
    • For angles less than about 15º, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.
    • A brief introduction to pendulums (both ideal and physical) for calculus-based physics students from the standpoint of simple harmonic motion.

  • Applications of Second-Order Differential Equations

    • 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.
    • The equation of motion is given as:
    • In real oscillators, friction (or damping) slows the motion of the system.
    • Including this additional term, the equation of motion is given as:
    • A solution of damped harmonic oscillator.

  • 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

  • Harmonic Syntax: Prolongation

    • To establish, or trigger, a harmonic functional zone, a fixed scale degree must appear in the bass.
    • The bass note of a passing chord will fill in the third with stepwise motion.
    • The melody will also often contain passing motion.
    • Thus instead of a passing motion of T1 D2 T3, a substitution pattern in the bass would produce T1 D4 T3.
    • Likewise, a divider chord takes a large leap between bass notes in a change-of-bass prolongation (or a simple octave leap in the bass) and divides it into two smaller leaps.

  • Sonata Form: Exposition Types

    • The bass note of a passing chord will fill in the third with stepwise motion.
    • The melody will also often contain passing motion.
    • In harmonic writing, the same effect is obtained by an incomplete neighbor chord.
    • Likewise, a divider chord takes a large leap between bass notes in a change-of-bass prolongation (or a simple octave leap in the bass) and divides it into two smaller leaps.
    • The bass line do–te–le–sol is harmonized by T1 D7 S6 D5.