Simple Harmonic Motion

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Simple Harmonic Motion

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.

Simple Harmonic Motion Equations

The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.

The velocity and acceleration are given by

The total energy for an undamped oscillator is the sum of its kinetic energy and potential energy, which is constant at

Energy transformation in periodic motion

Energy in Mass on Spring

The simple harmonic motion of a mass on a spring is an example of an energy transformation between potential energy and kinetic energy. In the example below, it is assumed that 2 joules of work has been done to set the mass in motion.

Mass on Spring Resonance

A mass on a spring has a single resonant frequency determined by its spring constant k and the mass m. Using Hooke's law and neglecting damping and the mass of the spring, Newton's second law gives the equation of motion:

The solution to this differential equation is of the form:

which when substituted into the motion equation gives:

Collecting terms gives B=mg/k, which is just the stretch of the spring by the weight, and the expression for the resonant vibrational frequency:

This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator.

source :- http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c4