Applications of

Simple harmonic motion is a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude.

It is characterized by its amplitude (which is always positive), its period which is the time for a single oscillation, its frequency which is the number of cycles per unit time, and its phase, which determines the starting point on the sine wave.

The period, and its inverse, the frequency, are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system.

Variation of acceleration with time

Displacement is given by:

Velocity is given by differentiating the above equation once:

Angular Frequency:

Differentiating once more gives us the acceleration:

On Simplifying:

Mass on a Mass on a pendulumpendulum

The above formula is used to express the time period of an ideal pendulum system:L is the length of the pendulum and g is the acceleration due to gravity.This shows that the period of oscillation is independent of both the amplitude and pendulum mass.

Mass on a spring

A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with: Even the formula on the

left can be used to calculate the Time period and hence shows that the period of oscillation is independent of both the amplitude and gravity.

The total energy is constant which is given by:

Given mass M attached to a spring pendulum with amplitude A with acceleration a:

k is the spring constantM is the massa is the accelerationA is the amplitude OR λ is the wavelengthf is the frequencyTs or Tp is the period of the spring or pendulum g is the acceleration due to gravityis the length of the pendulumEtot is the total energy

Definitions:Amplitude ( A ): The maximum distance that an object moves from its equilibrium position. Period ( T ): The time that it takes for an oscillator to execute one complete cycle of its motion. Frequency ( f ): The number of cycles (or oscillations) the object completes per unit time. Simple Harmonic Oscillator: Any object that oscillates about a stable equilibrium position and experiences a restoring force approximately described by Hooke's law.

Some useful and everyday examples are: a mass attached to a spring, a molecule inside a solid, a car stuck in a ditch being “rocked out”, a pendulum, an electron inside an atom.

simple harmonic motion is an effect

taking place throughout nature which has

many practical applications for human beings.

We must learn how to derive energy from simple harmonic motion as it can be used as a potential energy source especially when the world is surging toward an energy crisis.

Simple harmonic motion

spares no one -

ranging from the electron inside an atom

to the earth rotating around the sun.

2008-09