Sine Up to Learn All About Simple Harmonic Motion Functions

Photo Credit: Zátonyi Sándor/Wikimedia Commons

By Andrew Bennett

In this blog post, you will learn how to model an object's position as it undergoes simple harmonic motion using sine or cosine functions.

Simple Harmonic Motion Refresher

Simple harmonic motion is defined as periodic motion caused by a restoring force that is proportional to the object's displacement from equilibrium. This definition might seem arbitrarily complicated, but there are good reasons for separating this type of motion. For this post, it is important to note that the position, velocity, and acceleration of any object undergoing simple harmonic motion can be described using the sine and cosine functions. We might not be able to model other types of repeated motions with simple equations like this.

Sine Function for SHM

A general form frequently used here for the sine function is:

x(t) = A*sin (ꞷt)

Where:

x = the position at some moment in time
A = the amplitude of the motion
ꞷ (omega) = the angular frequency of the motion
t = the time

YouTube Screenshot (https://www.youtube.com/watch?v=e1b5Dkk8cYc)

Note that there is sometimes a "plus phi (+ɸ)" inside the parenthesis. This shifts the whole graph left or right. Since we can decide when to start our time measurements, we'll just choose to have zero time correspond with a moment when the oscillator is passing through equilibrium and moving in the positive direction.

Everything in the video below is done using the sine function but works equally well with cosine. The only difference is the starting position. If your oscillator is starting at its maximum displacement at time zero (and you don't have the option to change the meaning of time zero), cosine would be a logical choice.

How to Adjust the SHM Equation

In the video below, I'll show you how to determine the values for the coefficients to tailor this equation to the oscillator you are dealing with. Once you have an equation like this, you can do all sorts of nifty tricks with it, such as taking the derivative with respect to time to find a velocity function and once more to find the acceleration function.

You could also substitute these equations into some of the energy equations. For example, you can replace the speed term in the kinetic energy equation to get an equation for kinetic energy as a function of time.

If viewing via email, click here to watch the video.

Swing by My YouTube Channel for More Videos

You can subscribe to my channel to receive alerts when a new video is available. While you're at it, please subscribe to this blog, as well. Thank you!

Comments

Why Do They Do That? The Physics Behind 3 Famous Animal Behaviors

By Amber Bennett Have you ever wondered about the origins of the phrase, "get your ducks in a row"? What about how the "doggy paddle" works or why squirrels are such amazing jumpers? The answers to all of these questions have one thing in common: physics. Keep reading to learn about the science behind these three famous animal behaviors. How Do Squirrels Leap from Branch to Branch? A squirrel leaps through the air. Photo Credit: caroline legg , CC BY 2.0 , via Wikimedia Commons The death-defying acrobatic maneuvers performed by squirrels look a lot like parkour tricks. But how do they know how to land tricky jumps between bendy tree branches that move with the wind? Researchers say it's a combination of learned behavior (practice makes perfect) and inherited adaptations. To see what determines how far a squirrel is willing to jump and how the leaps are timed, researchers set up an obstacle course in an artificial forest. Then, they used peanuts to encoura

3 Easy Science Experiments to Do with Your Kids

By Amber Bennett Strong STEM (science, technology, engineering, and math) skills set kids up to succeed both in school and later in life. Encouraging them to be interested in these subjects is easier than you might think, especially if you start when they're young. Activities that encourage children to be curious and creative will spark a love of learning. Plus, science experiments and other hands-on STEM projects are a fun way to bond with your kids. Here are three easy science experiments that preschoolers and early elementary students will enjoy. 1. Leakproof Plastic Bag A (age 5) doing the leakproof plastic bag experiment. Our preschooler loves to do any experiment that involves water. I can give her a couple of beakers, and she'll happily pour water back and forth between them, especially if I put a few drops of food coloring in each beaker so she can mix the colors. Another great activity that involves water is the leakproof plastic bag. All you need are: Large plastic

How Virtual Currency Mining Is Hurting the Environment - Science in the News

Datacenters used for cryptocurrency mining use a lot of energy. By Amber Bennett Although cryptocurrencies are virtual, they are having a tangible impact on our environment. Research shows that cryptocurrencies actually are worse for the earth than other types of money. So, why is that? Simply put, it's because of the resources needed to run the supercomputers that mine Bitcoin and other virtual currencies. Energy Use The datacenters used for mining use large amounts of electricity. For example, the global Bitcoin network uses more energy than entire countries . In fact, according to the University of Cambridge Bitcoin Electricity Consumption Index , the network's power demand is the same as more than 14 million typical American homes. Most of the energy used in the datacenters comes from fossil fuels instead of renewable resources. Plus, datacenters' high energy use can destabilize the electric grid. Materials Mining Another way that cryptocurrency impacts the environme

Bennett Science Blog

Search This Blog