Oscillations occur all around you - anything that repeats could be considered an oscillation. There are obvious macroscopic oscillations, such as a mass connected to a spring (shocks on car) or a simple pendulum (grandfather clock), but oscillations are actually pervasive at all scales. The binding between two molecules acts very similar to if you had connected those molecules by a tiny spring. All matter is vibrating, and solids vibrate like little masses connected by little springs. The greater the temperature, the greater the energy and amplitude of the vibrations - the molecules wiggle more. On the celestial scale the moon orbits the Earth, which in turn orbits the sun. Then the whole solar system orbits the black hole at the center of our galaxy. Theorists and philosophers have even predicted The Big Bang is followed by a Big Crunch that is then followed by another big bang and the whole process repeats - this viewpoint is not in particular favor these days but give it more time and it may come back, kind of like an oscillation.

  --Forces and Energy--

All classic oscillations have a few things in common. First, when the system is displaced from its current state, there is a Restoring Force that drives it back. Think of a spring stretched or compressed, it has a spring force driving it back towards its relaxed length. Second, the potential energy (PE) function (U) of the system must have a Potential Energy Well. A PE well is a point where the potential energy increases in all directions. There are local and global wells, which refer the size scale in question. For example, water in my cup on my desk is in a local gravitational PE well but if it splashes out it could fall completely off the desk onto the floor, which may be considered a more global well. If you recall for a spring the potential energy is $\frac{1}{2}k \Delta x^2$, which is a quadratic function centered around $\Delta x = 0$. This is a potential energy well that increases for both positive and negative values of $\Delta x$, as shown below.

All oscillators share in common a restoring force and a potential energy well. What differentiates harmonic oscillators from other, non-harmonic oscillators, is the functional form of these forces and potential energy functions. A system will undergo simple harmonic motion if the restoring force is linearly dependent on the displacement from equilibrium.

$Force \propto x$

This means if the displacement is doubled, than so is the restoring force. A system will also undergo simple harmonic motion if the potential energy function that arises from the restoring force is quadratic with respect to displacement from equilibrium.

$Potential Energy \propto x^2$

This means that if the displacement doubles, the potential energy increases by a factor of four. These two features are actually dependent on each other and if the restoring force is linear, then the PE is quadratic (and visa versa).

Another important feature of a simple harmonic oscillator is that the period of the motion is independent of the energy. Consider a spring with a mass oscillating. As the system loses energy and the amplitude of the oscillation decreases but the time it takes for one cycle will remain constant.

  --Position, Velocity, Acceleration--

All oscillators display repetitious behavior in their kinematic equations (position, velocity, and acceleration), but Simple Harmonic Oscillators display distinctively sinusoidal behavior.

The exact form of the sinusodial equation depends on what are called the initial conditions, which is the state of the system at time equals zero. At t = 0 in the plots above, the system's position is a positive maximum, the velocity is zero, and the acceleration is at a negative maximum. If you were to hang a mass from a spring and release it from rest from its maximum height (assumed up is positive direction), it would ocsilate about and equilibrium position and could be described by the functions above (assuming no energy loss).

 --Omega and Uniform Circular Motion--

The omega found in the above equations is the same one used for angular velocity in Uniform Circular Motion (UCM). It's related to the period is the same way as well.

$\omega = \frac{2\pi}{T}$,   where T = period

To see the connection, the animation below shows the projection on the y-axis of an object in UCM. It takes the same amount of time for a complete oscillation and thus they each have the same $\omega$.

Pre-lecture Videos

• SHM features (6min)

SHM features (6min)

• SHM 2nd law and sinusodial (6min)

SHM 2nd law and sinusodial (6min)

• SHM position function (5min)

SHM position function (5min)

• SHM velocity function (7min)

SHM velocity function (7min)

• SHM acceleration function (3min)

SHM acceleration function (3min)

Supplemental but suggested

• Energy review, wells, forces of springs (22min)

Energy review, wells, forces of springs (22min)

• SHM summary of relationships (4min) **

SHM summary of relationships (4min) **

• SHM equations and variables example (9min) **

SHM equations and variables example (9min) **

Lecture Notes | (PDF)(OneNote)

Note: At times Simple Harmonic Motion will be abbreviated as SHM.

Section 16.2 from the Openstax text covers Hooke's Law, 16.3 covers Period and Frequency in Harmonic Oscillators, 16.4 covers Simple Harmonic Oscillators as a whole and links simple harmonic motion, to waves.

The text from Boundless covers SHM in the sections Simple Harmonic Motion, as well as the following chapter Simple Harmonic Motion and Uniform Circular Motion

This page also covers SHM in a much more brief matter than the texts above.

Note: This resource travels into the land of Springs and Pendulums, save that for later.

Hyperphysics' quick page on simple harmonic motion

University of Winnipeg provides a nice quick reference page for SHM as well

Other Resources

This link will take you to the repository of other content related resources.

This video is a good lecture on Simple Harmonic Motion, towards the end he begins to talk about springs and pendulums, you can save that for the next section.

https://www.youtube.com/embed/DVtyRt0CSSw?rel=0

Note: In the video, the lecturer talks a few times about "taking derivatives", whenever he mentions derivatives just think about slopes. When he says "we need to differentiate twice with respect to time" He is talking about the function for position, which if you find the slope of a line on a position vs time graph you get velocity; and if you take the slope of that line on a velocity vs time graph, you will have the acceleration!

Doc Schuster gives a we rounded (17 minute) lecture on simple harmonic motion, including various example problems. Again, when derivatives are brought up, just think slope.

Crash Course Physics discusses Simple Harmonic Motion and the algebraic math used to describe it.

Other Resources

This link will take you to the repository of other content related resources.

The simple pendulum is a special physical case that represents simple harmonic motion. This PhET simulation you can explore the effects of mass, length of cable and gravity on a simple pendulum, which is covered in more detail in our next section, Springs and Pendulums.

For additional simulations on this subject, visit the simulations repository.

For additional demos involving this subject, visit the demo repository

1.  The period of oscillation for on object is observed to be $10$ seconds. 

     (a)  What is the frequency?

     (b)  If the frequency is doubled, what is the new period?

2.  Which of the following 1-D forces gives rise to simple harmonic oscillation.

     (a)  $F = - 8.6 \, y$

     (b)  $F = 7.5 x$

     (c)  $F = - 3.09 \, \theta$

     (d)  $F = -\frac{1}{1} \, x^2$

3.  The plot below shows a potential energy vs position for an object moving in 1-D.

     (a) identify all of the unstable equilibrium location.

     (b) identify all of the stable equilibriums locations.

     (c) If an object is placed at each stable equilibrium and displaced slightly from its equilibrium position, which locations would mostly closely resemble simple harmonic motion?

Click HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

BoxSand practice problems

Conceptual problems

KC's multiple select problems

KC's quantitative problems

Recommended example practice problems 

The following sets are from the OpenStax text. The problems are located after the subject material.

• Period and Frequency Exercies, Website Link
• Simple Harmonic Motion Exercises, Website Link
• Uniform Circular Motion Exercises, Website Link

This problem set is 20 questions long, answer key is on the last page, PDF Link

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.