Masses connected to ideal springs and simple pendulums are examples of Simple Harmonic Oscillators. When set into motion the masses will oscillate back and forth in a predictable sinusoidal way. These two systems provide excellent case studies in SHO.

Check out the mesmerizing Double Pendulum

https://www.youtube.com/watch?v=hXOEoH5q3Hw

## 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

## Required Videos

## Suggested Supplemental Videos

### Learning Objectives

## Summary

Summary

## Atomistic Goals

Students will be able to...

### BoxSand Introduction

## Oscillations | Springs and Pendulums

A Simple Harmonic Oscillator is a system where the restoring force acting on the object is linear with respect to displacement and it's potential energy is quadratic with respect to displacement. Masses connected to springs and simple pendulums behave this way and thus provide excellent case studies into the features of SHO.

--**Springs**--

Recall the force acting on an object fixed to the end of a spring is $|F^s| = k |\Delta x|$, where x is the displacement from equilibrium and k is the spring constant. Also recall that the potential energy is $U^s=\frac{1}{2}kx^2$. Here the displacement variable is a position measured by distance. The kinematic equations describing the motion are sinusodial, as expected for all SHO. In general, all SHO have an angular velocity $\omega=\frac{2 \pi}{T}$, where T is the period. For a spring specific system, there is also the useful relationship below.

Here you see $\omega$ (and thus T) is independent of the energy of the system. This is an expected feature among SHO where the period is independent of the amplitude, and thus energy, of the oscillation.

--**Simple Pendulums**--

A mass connected to a light string - a simple pendulum - that oscillates at small angles ($\theta < 10^0$), the net force of gravity and the string act as approximately a linear restoring force. This means the pendulum behaves like a SHO. The natural variable to track the position of the pendulum is the angle $\theta$, measured with respect to the vertical. With this as the displacement variable, the motion follows a sinusoidal form. An example for a pendulum released from rest from a positive maximum position ($\theta_{max}$) at t = 0, this would look like the following,

$\theta (t) = \theta_{max} cos(\omega t)$, where $\omega$ is the angular velocity.

All SHO have the relationship that $\omega = \frac{2 \pi}{T}$, where T is the period, but pendulums also have a useful relationship between $\omega$ and the length of the pendulum (l), and the gravitational field (g) they pendulum resides in.

Here you see $\omega$ (and thus T) is still independent of the energy of the system but also independent of the mass connected to the object. If the pendulum is raised to angles greater than 10 or 15 degrees, it no longer behaves like a SHO and the motion requires additional considerations.

## Key Equations and Infographics

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

### OpenStax Reading

OpenStax Section 16.4 | The Simple Pendulum

OpenStax Section 16.5 | Energy and the Simple Harmonic Osciallator

OpenStax Section 16.6 | Uniform Circular Motion and Simple Harmonic Motion

## Additional Study Resources

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

### YouTube Videos

Yau-Jong Twu gives a great lecture on the mass-spring system.

https://www.youtube.com/embed/3pJrr6PdrWE?rel=0

AK Lectures gives a detailed lecture on the simple pendulum

https://www.youtube.com/embed/LbQuKZD3svI?rel=0

Brightstorm covers Hooke's Law and does a few examples.

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

### Other Resources

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

### Simulations

This PhET simulation you can explore the effects of mass, length of cable and gravity on a simple pendulum,

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

Boundless has several short sections that pertain to Springs and Pendulums. This link takes you to a table of contents where the relevent sections are titled: Hooke's Law, Period of Mass on a Spring, Waves and Vibrations, The Simple Pendulum, and the Physical Pendulum.

From The Physics Classroom we have,

Motion of a mass on a spring, | Pendulums |

Here are the Hyperphysics sections for Hooke's Law, and Pendulums,

Hooke's Law | Pendulum |

The resource from Boston University referenced in our section on Simple Harmonic Motion also contains spring and pendulum information.

### 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

The two main model we use for simple harmonic motion are the pendulum and the spring. Important differences between them are the source of the restoring force. For the spring, it comes naturally from Hooke's law F=-kx. For the pendulum it comes from gravity's interaction with the tethered mass. Springs we discussed in detail last lecture, so let's talk about the assumptions of a pendulum.

In this class, we will examine simple pendulum, which assume a massless string connected to point mass (meaning all of the mass is confined to one specific point at the end of the pendulum). Important to realize there are a few other types of pendulums that we won't be studying in detail. We will also be making the liberal use of the small angle approximation. The small angle approximation is where when we have a small angle (meaning our pendulum isn't displaced far from the equilibrium position), we can make assumptions about trig functions that simplify things, the common trig functions reduce as shown below:

When $\theta$ is small

$\sin(\theta) \approx \theta$

$\cos(\theta) \approx 1$

$\tan(\theta) \approx \theta$

It is very very common to be given a problem where you will need to remember to use the small angle approximation in order to be able to solve a equation (even on exams). Many problems will not state explicitly that the small angle approximation can be used, instead, they will often use language that states some relative difference in lengths. If you notice language like "a distance much larger than the other length" or "very far away" think about what this means about the angle when you draw the picture, as this is usually a physics problem's way of justifying the use of the small angle approximation. The reason we don't directly state it for you is because we want you to be able to apply the reasoning of this approximation yourself and be able to recognize it in different physical contexts.

### Checklist

1. Read and re-read the whole problem carefully.

2. Spring and pendulum oscillation problems can usually be split into 3 types based off of the initial information given; graphical, mathematical, and pictorial. There could then be overlap of the two types (e.g. from a graph find the period...).

a. For graphical problems:

i. Identify and make note of the axes labels:

a. If the axes labels are potential energy vs time, identify the stable equilibrium. For springs and pendulums with small displacements about the equilibrum angle, this will plot will be a quadratic function, indicating that the motion is SHM.

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.

c. With this information, we can use known relationships to find the max (angular)/velocity, max (angular)/acceleration, and total energy.

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

a. Notice that the force will be proportional to the negative of displacement, this indications the motion is SHM for springs and pendulums about small angles.

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. The restoring force for both springs and pendulums will be proportional to the displacement in the opposite direction, indicating that 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 negative sign in Hooke's law doesn't mean the force due to springs always takes a negative value - the force will be positive or negative depending on where in the cycle you are looking.
- Extension/compression between $x_{max}$ and $x_{eq}$ is one quarter of a full cycle, not a half of a cycle.
- The period is independent of the amplitude of the oscillation for both springs and pendulums
- The frequency of a pendulm is proportional to $\sqrt{\frac{g}{l}}$, but only under the small angle approximation. All pendulum physics we study in this series will make use of the small angle approximation.
- The amplitude is measured from the equilibrium position, it is not the difference between the max and min values.
- Consider a pendulum in SHM. The angular frequency we use to describe this oscillatory motion $\omega$, is not the same as the angular velocity studied in earlier topics like rotational kinematics. To help differentiate between the two, use $\Omega$ to describe the angular velocity of the pendulum.
- Cosine and sine differ by a phase of $\frac{\pi}{2}$, not by a phase of $\pi$.

### Pro Tips

- Lots of spring problems can be solved very simply using energy. For example, what if you wanted to know how far a spring would compress ($\Delta x$) after a block slides into it? An energy analysis quickly leads to $m v^2 = k\Delta x^2$, which can be easily solved for $\Delta x$.
- If a problem asks for the time it takes for a spring to compress from equilibrium, a great shortcut is to use the following reasoning: since a compression is one-quarter of a period of simple harmonic motion - (1) compression, (2) motion back to equilibrium, (3) extension, (4) motion back to equilibrium - the time it takes for just compression is $t = \frac{1}{4} T$

### Multiple Representations

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

### Physical

The pendulum mass system is best modele with a picture that can be easily translated to a free body diagram.

### Mathematical

$E=K+UConst.$

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

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

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

Springs:

$\omega = \sqrt{\frac{k}{m}}$

Pendulums:

$\omega = \sqrt{\frac{g}{l}}$

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

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

### Graphical

The position, velocity, and aceleration graphs for spring and pendulums behave similarly to non oscilatory motionn graphs. The relationships between position, velocity, aceleration 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 aceleration at corresponding points in time.

### Descriptive

### Experimental

One key experiment is finding the period of oscillation of a pendulum. Until you have worked through the physics, it would seem that the size of the mass on the end of the penulum would affect the period of oscillation. However, we know from the mathematical representation that this is not true. In the following demonstration, we have a bunch of different length strings with the same size mass on the end of each string. As the mass's are released, observe each pendulum oscillate with a different frequency. Since all the masses are equal, it is clear that the period of oscillation is dependent on the length rather than the mass.

## Practice

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

### Fundamental examples

1. The displacement from equilibrium is plotted vs time for a spring mass system underging simple harmonic oscillation. The mass is $0.5 \, kg$.

Determine the...

(a) ...amplitude.

(b) ...period.

(c) ...frequency.

(d) ...maximum speed that the oscillator reaches. Identify the locations on the plot with a circle.

(e) ...maximum acceleration of the oscillator? Idensity the locations on the plot with a square.

(f) ...spring constant.

(g) ...total energy.

(h) Write an equation for this displacement as a function of time.

2. A simple pendulum of mass $0.25 \, kg$ undergoes simple harmonic motion as described by the equation below.

$\theta (t) = \frac{\pi}{18} \, cos\left( \frac{\pi}{2} \, t \right)$

Determine the...

(a) ...amplitude.

(b) ...period.

(c) ...frequency.

(d) ...maximum angular speed that the oscillator reaches.

(e) ...maximum angular acceleration of the oscillator?

(f) ...length of the pendulum.

(g) ...total energy.

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

BoxSand's multiple select problems

BoxSand's quantitative problems

Recommended example practice problems

- OpenStax, Period and Frequency also linked in SHM section, it's a good idea to get more practice!
- Hooke's Law SHM, Website Link
- Period and Frequency, Website Link
- The Simple Pendulum, Website Link

- 20 problem set on springs and simple harmonic oscilators. PDF Link

For additional practice problems and worked examples, visit the link below.

Use the Botafumeiro in Santiago de Compstela to estimate the height of the cathedral.

https://www.youtube.com/watch?v=45fNNo77WUU

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.