Summarizing Simple Harmonic Motion

Summary

• Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
• Maximum displacement is the amplitude \(X\). The period \(T\) and frequency \(f\) of a simple harmonic oscillator are given by

  \(T=2\pi \sqrt{\cfrac{m}{k}}\) and \(f=\cfrac{1}{2\pi }\sqrt{\cfrac{k}{m}}\), where \(m\) is the mass of the system.

• Displacement in simple harmonic motion as a function of time is given by \(x(t)=X\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\cfrac{2\pi t}{T}.\)
• The velocity is given by \(v(t)=-{v}_{\text{max}}\text{sin}\cfrac{2\pi \text{t}}{T}\), where \({v}_{\text{max}}=\sqrt{k/m}X\).
• The acceleration is found to be \(a(t)=-\cfrac{\mathrm{kX}}{m}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\cfrac{2\pi t}{T}.\)

Glossary

amplitude

the maximum displacement from the equilibrium position of an object oscillating around the equilibrium position

simple harmonic motion

the oscillatory motion in a system where the net force can be described by Hooke’s law

simple harmonic oscillator

a device that implements Hooke’s law, such as a mass that is attached to a spring, with the other end of the spring being connected to a rigid support such as a wall