Oscillations are one of the most prevalent phenomena in our universe. From a swinging tree branch, to molecular vibrations, even to the orbits of celestial bodies, repetitive events happen everywhere in nature. Simple harmonic oscillators (SHO) are specific type of oscillation where the motion can be described by sinusoidal functions and the time per revolution (period) is independent of the amplitude of the vibration.

Here is a quick introduction of Harmonic Motion

https://www.youtube.com/watch?v=9KMpE6BGz5Q

## Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

### BoxSand Videos

### Learning Objectives

## Summary

Summary

## Atomistic Goals

Students will be able to...

### BoxSand Introduction

## Oscillations | Simple Harmonic Oscillations

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

*. 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.*

**Potential Energy Well**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 $\omega$.

Or check out both the horizontal and vertical projections at the same time. Notice they are simply out of phase with each other in the cycle.

*Images thanks to: Wikimedia Commons

## Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

### OpenStax Reading

OpenStax Section 16.1 | Hooke’s Law: Stress and Strain Revisited

OpenStax Section 16.2 | Period and Frequency in Oscillations

OpenStax Section 16.3 | Simple Harmonic Motion: A Special Periodic Motion

## Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

### YouTube Videos

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.

https://www.youtube.com/watch?v=iNDRQnhIMK8

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

https://www.youtube.com/watch?v=jxstE6A_CYQ?rel=0

### Other Resources

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

### Simulations

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.

### Demos

### History

Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Physics Fun

### Other Resources

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.

## Resource Repository

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

## Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

### Assumptions

While simple harmonic motion is one of the most widely applicable systems in all of physics to natural phenomena, it is still a highly idealized system. The best way to conceptualize the model is with a horizontal spring with a mass on the end. In simple harmonic motion, we assume the only force present (after the object is set into motion) is the restoring for, which for a spring is Hooke's Law F=-kx. Where there are two really important subtle things in that equation. One is that the x is the distance away from the equilibrium position. One of the first things you should do in a simple harmonic motion problem is to determine the equilibrium position (it is always exactly halfway between the minimum and maximum position). The other is the negative sign, which means the force will always be pointing in the opposite direction of the way in which the mass is displaced. This is the definition of the restoring force. Because we have to displace FROM EQULIBRIUM, our displacement as a vector will point away from equilibrium, and the negative sign makes the force ALWAYS point towards equilibrium (and be 0 when the mass is at equilibrium). Once more, because this is the only force present, meaning no friction or anything we usually think about in the real word, this oscillatory movement will continue forever. This won't be true anymore when we look at driven and damped oscillations, but for all simple harmonic motion examples, this is true.

We took Newton's 1st law as an object in motion will stay in motion unless acted upon by an outside force and... well.... this is that, other than our force keeps the object moving in a oscillatory way. However, the in simple harmonic motion, we never gain or lose any energy, all kinetic energy from the motion is converted to potential energy as the mass reaches maximum displacement and it comes to a stop, and the total energy in the system is always constant. Understand this as a "toy model," one that doesn't have all the complications of reality, but serves as an excellent idealized model for oscillatory motion, and believe me, it is one of the most widely used models in all of physics. It turns out one of the most mind blowing results to come out of quantum mechanics is that all matter is made up of waves more or less, and at this fundamental level, most of those waves can be modeled as simple harmonic oscillators to a great degree of accuracy. Check out this Wikipedia page for more if you're interested: https://en.wikipedia.org/wiki/Matter_wave

### Checklist

i. Identify and make note of the axes labels:

a. If the axes labels are potential energy vs time, identify stable and unstable equilibrium locations. If displaced from a stable equilibrium location, oscillations will occur.

i. If the stable equilibrium locations have the same shape as a quadratic funtion, then the oscillations about this equilibrium can be approximated as simple harmonic oscillations.

b. If the axes labels are displacemnt vs time, identify the amplitude, and the period directly from the graph. The angular frequency can then be determined from the wavelength $\omega = \frac{2 \, \pi}{T}$ .

i. Look at the initial conditions (at $t = 0$ ) to identify if the oscillation can be written as a sine or cosine function.

b. For mathematical problems:

i. If the mathematical equation is a displacement as a function of time:

a. Identify the amplidue from the given equation.

b. Identify the angular frequency from the equation; relate this angular frequency to the period of oscillation.

ii. If the mathemaical equation is a force as a function of displacement:

a. Identify if the force is proportional to the displacement. If so, check if the force acts in the opposite direction of displacement (there will be a negative sign in the equation). If both conditions are met, then this force is describing a SHO.

b. If there are any velocity dependent forces that also act in the opposite direction of displacement along with a linear displacement force that acts in the opposite direction of motion, then the equation is describing a damped oscillation.

c. For pictorial problems:

i. Visualize the scenario. Mentally try to understand what the object(s) is(are) doing and what forces are acting on it (them).

ii. Use Newton's 2nd law to write an equation of motion for the system.

iii. Determine if there is a restoring force that is proportional to displacement. If this is the only force, then the motion is SHM.

iv. Determine if there is a damping force along with the restoring force. If this is the case, then the motion is damped harmonic motion.

v. Rewrite your Newton's law translation into a form of the equation of motion. $ \left( a_x \rightarrow \frac{\Delta^2 x}{\Delta t^2} \right)$. With this new form, identify the angular frequency, and if damped, identify the time constant.

3. Evaluate your answer, make sure units are correct and the results are within reason.

### Misconceptions & Mistakes

- The argument that goes into a sine or cosine (any trig fucntion really), does not have units. The evaluation of this argument also does not have units. For example, with $sin(\omega \,t)$, the argument is "$\omega \, t$" which when multiplied together the units cancel, and when we take the sin of that value it returns a unitless value.
- All oscillations are not simple harmonic oscillations. Simple harmonic oscillations are a special type of oscillation where the restoring force is linear with respect to the displacement from equilibrium; this is the same as saying that the potential energy plot vs displacement is quadratic.
- The amplitude is measured from the equilibrium position, it is not the difference between the max and min values.
- Although we are using circular motion variables and quantities (e.g. period, angular frequency, etc...), the underlying physical phenomena we are describing need not be circular in nature; for example a mass on a spring is linear motion yet we can still describe this motion with quantities such as period and angular frequency.

### Pro Tips

- Sine and cosine are very similar functions - they only differ by a phase. If you look at a plot of sine versus cosine, you can see that if you shift one of the plots by $\frac{\pi}{2}$, they become the same function.
- The slope of a position versus time plot is its velocity; the slope of a velocity versus time plot is its acceleration. The slope of a sine function is cosine. So if you have a function that describes the position of an object that begins at its max amplitude $x = x_max sin(\omega t)$ , you quickly know that the function to describe its velocity is $v_x = v_max cos( \omega t)$. Understanding the relationship in terms of slopes instead of just memorizing off an equation sheet will help you answer qualitative questions on exams.
- Sine and cosine functions come up over and over, so its worth spending some time practicing now to get familiar with them. In particular, it is often helpful to understand the limits - that $cos(0) = cos(2\pi) = 1$, etc. Some find the unit circle to be an incredibly helpful visual tool for remembering these limits - ignore all of the $\frac{\sqrt{2}}{2}$ stuff, and just memorize the values of sin and cosine at $0, \frac{\pi}{2},\pi, \frac{3\pi}{2}$, and $2\pi$.

### Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

### Physical

A spring mass system that oscilates back and forth. There are several important positions that the objects oscilates between. The max and min position points are related to the positions whre the veloicity is zero at the turning points. In addition, at the equilibrium position the velocity is maximum.

### Mathematical

$E=K+U=Const.$

$\nu_{max} = \omega x_{max}$

$a_{max} = \omega^{2} x_{max}$

$\omega = 2 \pi f = \frac{2 \pi}{T}$

$x(t) = \pm x_{max}\frac{\sin \text{or}}{\cos} (\omega t)$

$\nu (t) =\pm \nu_{max} \frac{\sin \text{or}}{\cos} (\omega t)$

$a (t) =\pm a_{max} \frac{\sin \text{or}}{\cos} (\omega t)$

Energy ($E$), Velocity ($V$), Angular frequency ($\omega$), Distance ($x$), Amplitude ($A$), Spring Constant ($k$),

Mass ($m$), Gravity ($g$), Length ($l$), Period ($T$)

### Graphical

The position, velocity, and acceleration graphs for simple harmonic oscillators behave similarly to non oscillatory motion graphs. The relationships between position, velocity, acceleration still hold. The slope of the position graph corresponds to the value of the velocity at corresponding points in time. In turn, the slope of velocity corresponds to the values of acceleration at corresponding points in time.

### Descriptive

Simple harmonic oscillators are systems whose motion is described by a linear restoring force.

### Experimental

## Practice

Use the practice problem sets below to strengthen your knowledge of this topic.

### Fundamental examples

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.

### Practice Problems

Conceptual 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.