Modeling Damped Harmonic Motion

In this activity, you'll use a video measurement and analysis tool called Tracker.Tracker enables you to create two different types of mathematical models:analytical and dynamic.In an analytical model, you can enter mathematical expressions forxandypositions as a function of time.That's sometimes useful, but from a physics perspective, a dynamic model is much more flexible and powerful.

With a dynamic model, you can set the initial conditions for a particular system (initial positions and velocities); then you can mathematically define any forces acting on that system.Once those are set up, the model becomes like an object in space, responding to the forces you've imposed on it.It might continue moving forever, if that's what the forces would do to an object in real life.By visually matching a marker for your model to the real motion on the video, you can define and refine a mathematical model for a wide range of real-world situations.

In this activity, you'll use Tracker to observe the displacement, velocity, and acceleration of a mass on a spring.Tracker is a free scientific tool available through the Open Source Physics Project.

Preparation - Creating a Dynamic Particle Model If you've never built a dynamic model before using Tracker, you'll need to prepare a bit and find out where you can get help in creating your models.In Tracker, you can always access illustrated Help.For this project, you're going to need to check out the Tracker Help instructions for creating a dynamic model.

In Tracker, just click on theHelpmenu, then, clickTrackerHelp.In the Tracker Help window, clickParticleModels.This short, illustrated document tells you everything you need to know to create a dynamic particle model in Tracker.You can print this Help document, but it is available from Tracker anytime you need to refer to it.

Instructions - Building your Dynamic Model Start your activity by opening this Tracker experiment:Spring Wars model (https://contentstore.ple.platoweb.com/content/sharedmedia/Tracker/applications/spring-wars/spring-wars.html)

Click playto watch the video.The other video controls allow you to rewindthe video or step forwardor backwardone frame at a time.

You've seen the motion of a mass on a spring where the motion is largely "undamped"; that is, it continues for a very long time at about the same amplitude.There are two common ways to slow down an object in harmonic motion:

• kinetic friction - directly proportional to weight and opposite the direction of velocity
• spring damping - directly proportional to velocity and opposite the direction of velocity

Both of these damping forces are at work in this video, but the frictional force is by far the most significant here.

A dynamic model is already started for you in this file.Follow the two steps in the screen captures below to open the model setup and begin your modeling work.

1. Select the point massModel. (from the blue pull-down control in the top menu)
2. Open theModelBuilder. (from the "Model" mass pull-down menu)
3. Once you've opened the Model Builder screen, use theTrackerHelpdocumentation to begin filling out the basic information about this motion.You only have complete data table information starting at timet= 0.00 seconds, so use that as your initial time.Enter data in each of the three sections:
   • parameters -Set m = 0.250 kilograms.Add a parameterkfor the spring constant and a parametermkfor the coefficient of kinetic friction.
   • initial values - From the data table, enter those that apply to thisx-direction motion:t,x, andvx.
   • force functions - It will be easier and cleaner to enter the force function as a sum of two forces, the spring force and the frictional force:fx:Fs+Ff.(Notice that this section turns red after you enter the force functions, since you have not definedFsandFfyet.)
   • Add a new force function forFs, the spring force.Note that the equilibrium point is atx= 0, so this force law should be pretty easy for you.
   • Add a new force function forFf, the frictional force.
4. Finally, to make your model work, you'll have to find values for the spring constantkand the coefficient of friction_k.You'll just have to play around with_k, but you can get close to the spring constant value by looking at some of your data.The first question below will help you with that.

Part A

Calculate the spring constant.

• Look at the graph ofxvs.tgraph.(You may want to double-click on it to examine it in theDataToolview.)
• From the graph, determine the periodT.
• The first video frame lists the mass.
• Use the equation below to solve for the spring constantk.Show your work below.

T=2 Root m/k

Part B

Enter into the table your calculated value for the spring constant, then play with different values of mk until you get a close match to the motion.(Note:It will never be perfect.Remember that there are two kinds of spring damping.Both are at work here, but we are not going to model both.) Once you're satisfied with your model, record your model values in the table below.

Variables	Expression
parameter: m
parameter: k
parameter: mk
initial value: t
initial value: x
initial value: vx
fx
Fs
Ff

Part C

Describe how your model works compared to the actual spring motion.When are they close together?When are they not so close?

Part D

Now that you've had some experience with damped harmonic motion, describe two different real-world harmonic motions you've seenone with very little damping and the other with noticeable damping.Describe how those two motions differ and are the same.

Amusement Park Physics

Amusement parks provide many hours of fun, but did you know you can also practice your physics there? Yes, parks rides rely on physics to function. Let's look at rides that travel in a circular path, such as a roller coaster or carousel. These rides illustrate the concept of centripetal force. We will also consider the sensation of being pulled to the outside of these rides, which we call centrifugal force.

Part A

Why do we experience centripetal force and acceleration on a carousel?

Part B

Many people believe that it is centrifugal force that causes the sensation of being pulled to the outside on a carousel. What accounts for this common misconception?

Part C

Research how a centrifuge works. Relate the mass of an object to its centripetal force, and give examples of how centrifuges are used. Include at least two authoritative research sources to support your response. Record these sources at the end of your response.