Figure 12.1: A simple pendulum

Transcription

1 Chapter 12 A Simple Pendulum by Brian Patterson In this module you will use DIYModeling to build a simulation of a simple pendulum. The basic ideas can be extended to other types of pendulums, such as physical, spherical or torsional pendulums The Physics of a Pendulum Figure 12.1: A simple pendulum A simple pendulum consists of a point mass m swinging at the end of a massless string of length L, as shown in Figure The angle that the string makes with vertical at any instant is, and the motion is confined to a single vertical plane. It is straightforward to find the equations of motion using the rotational form of Newton s second law, 105

2 CHAPTER 12. A SIMPLE PENDULUM 106 X = I. The only torque ( ) acting on the pendulum is caused by gravity (we ignore any frictional losses) and can be written as = mgl sin, and the rotational inertia (I) for the point mass is I = ml 2. Putting these two equations together and writing the angular acceleration ( ) as the second time derivative of the angular position ( ) gives or mgl sin = ml 2 d2 dt 2 d 2 dt 2 = g sin. L In DIYModeling we express this second order di erential equation as a system of two first order di erential equations d dt d! dt =! = g L sin where! is the angular velocity of the pendulum in radians per second (and should not be confused with the angular frequency of oscillation). We will also use initial conditions (0) = 0!(0) =! 0

3 CHAPTER 12. A SIMPLE PENDULUM 107 Figure 12.2: The Project tab Note that because we are using DIYModeling, which solves these di erential equations numerically, students can study the motion of a simple pendulum without recourse to the small angle approximation, as is typically done in standard textbooks Building the Model and Simulation in DIYModeling Launch DIYModeling and open the QuickStart skeleton file pendulumtemplate.xml. See Section 1.2 on page 6. This skeleton file does not yet implement a working simulation you will need to modify it a bit to get it to work, as outlined below. (Note: if you want to examine a fully functional pendulum simulation, open the example file pendulum.xml and run it in the usual way.) In the Project tab (See Figure 12.2), you will see some basic information: a descriptive Model Name ( Simple Pendulum Template ), the World Time Units ( seconds ) and World Length Units ( meters ) used in the model, and Time Scale and Visual Scale factors (both set to 1 because no scaling is needed for this model). If you were creating this model completely from scratch, you would need to provide this information yourself.

4 CHAPTER 12. A SIMPLE PENDULUM Adding Components Figure 12.3: The finished simulation Looking ahead a bit, Figure 12.3 shows the final rendering of the pendulum model in DIYModeling. It includes a support structure to which the pendulum is attached, a length of string, and the pendulum mass (or bob ). Not explicitly shown, but also included as components are the camera and light source. To create these components in your model, click on the Components tab in the Model Editor window. In the large area on the right of the screen (see Figure 12.4) you will see that the camera and light source have already been provided in your model components, but you will need to add the other three components yourself. From the Component Libraries pane to the left of the screen, scroll down until you find the standard component Cube. Click and drag this component to the right side of the screen, where it will show up as a component named Cube1. You will use this component for the pendulum support. Note that, by double-clicking on Cube1, you can rename it as something a little more descriptive (e.g, Support ). Next, click on

5 CHAPTER 12. A SIMPLE PENDULUM 109 Figure 12.4: The Components tab the standard component Sphere in the Component Libraries pane and drag it to the right side of the screen, where it will show up as a component named Sphere1. You will use this component for the pendulum bob, so double-click on it and rename it as PendulumBob. Finally, click and drag the standard component Arrow from the Component Libraries to the right side of the screen. You will use this component as the pendulum string, so rename it as String. At this point your component set should resemble Figure Although you have now created all the required visual components for your model, they are neither correctly sized nor positioned (go ahead and run your model to see this for yourself). To fix this, click on the small button to the right of the Support component. A new region will open in the center of the screen labeled Component Attributes and Parameters, where you can specify both the size and position of the Support component. In the Size field (see Figure 12.6), enter [5, 1, 5] which changes the size of the Support component to 5 units in the x and z (horizontal) directions and 1 unit in the y (vertical) direction. Then change the Position field (see Figure 12.7) to [0, 0.5, 0]

6 CHAPTER 12. A SIMPLE PENDULUM 110 Figure 12.5: The Components tab with the new components which shifts the Support component vertically by 0.5 unit, so that its base is in the xz plane. In the same way, click on the small button to the right of the PendulumBob component, and change the Diameter field to [1, 1, 1] This sets the PendulumBob diameter to 1 unit. variable name In the Position field, enter the bobposition Finally, you will need to change the end points of the String so that it starts at the origin (i.e., at the pivot point) and ends at the center of the bob. Click on the small button to the right of the String component, and enter [0,0,0] in the ArrowTail field and enter the variable name

7 CHAPTER 12. A SIMPLE PENDULUM 111 Figure 12.6: The Support size parameter

8 CHAPTER 12. A SIMPLE PENDULUM 112 Figure 12.7: The Support position parameter

9 CHAPTER 12. A SIMPLE PENDULUM 113 Figure 12.8: Linking the PendulumBob position to the variable bobposition bobposition in the ArrowHead field. This variable name has not yet been defined, but we will use it later on to specify the three-dimensional position of the pendulum bob. You have now finished creating your model components. See Figure Specifying the Model Now that you ve selected your components, you will need to enter the mathematical model that controls their behavior. Click on the Model tab in the Model Editor. The first two lines of the model table have already been created for you (see Figure 12.9) and define the constants appearing in the problem: the acceleration due to gravity (g = 9.81 meters per second 2 ) and the pendulum length (which is set to length = 10 meters). Later we will add a slider control that will allow you to easily adjust the value of length, but for now we ll just keep length constant at 10 meters. In the next three lines of the model table, you will create the dynamical variables for

10 CHAPTER 12. A SIMPLE PENDULUM 114 Figure 12.9: The first two lines in the model the angle ( ), the angular velocity (!), and the position of the pendulum, as described below. To create a variable for the pendulum angle, click on the Add New button at the top of the table, and, in the new line that appears, enter theta in the Name column. Because is governed by a di erential equation, select di eq from the drop down menu in the How column. The pendulum angle is a simple scalar function, so you should leave the What column unchanged ( decimal ). Click in the Units column and select radians for the angular units, and enter a reasonable initial value for in the Initial Value column, say 0.5 radians (about 30 degrees). Finally, in the Expression column, enter the time derivative for, given by d dt =! That is, you should simply enter omega (the variable name you will use for angular velocity) in this cell. Check your inputs for theta against line 3 of of Figure To create a variable for the angular velocity, click the Add New button and, in the new line that appears, enter omega in the Name column. The values you enter in this row should parallel those for (or theta) di eq, decimal and radians/s, in the columns How, What, and Units, respectively. We ll release the pendulum from rest, so enter 0 as the Initial Value. Finally, in the Expression column, enter the time derivative for!, given by That is, you should enter d! dt = g L sin

11 CHAPTER 12. A SIMPLE PENDULUM 115 Figure 12.10: The model -(g/length) * sin(theta) Check your inputs for omega against line 4 of Figure To create a variable for the position of the pendulum bob, click the Add New button and, in the new line that appears, enter bobposition in the Name column. You will calculate the bob position as a simple function of the angle and length of the pendulum, so select function in the How column. The variable bobposition will be a three-dimensional vector, as it must specify the x, y and z components of the bob position, so select vector in the What column and enter 3 in the Dim column. Select meter for units. Finally, in the Expression column, enter the vector position of the bob, [sin(theta), -cos(theta), 0] * length Check your inputs for bobposition against line 5 of Figure Run your simulation in the usual way by selecting Simulate and Run from the Run pull-down menu in the Model Editor. If you ve made no mistakes, you should see your pendulum swinging back and forth beneath the support structure (see Figure 12.3). If your simulation doesn t run, check the Error tab in the Model Editor, fix any errors, and re-run the simulation. If you have problems troubleshooting your model, compare your work to fully implemented model pendulum.xml provided in the QuickStart Collection.

12 CHAPTER 12. A SIMPLE PENDULUM Adding Some Bells and Whistles To make the simulation more useful, try adding a couple of slider controls (see Chapter 9) for the pendulum length and initial angle, so that you can easily vary these parameters during the simulation. Briefly, To create a slider control for the pendulum length, go to the Components tab of the Model Editor. Select a slidercontrol standard component from the Component Libraries on the left side of the screen and drag it into your collection of pendulum components on the right side of the screen. Then double-click on the slider icon and rename it as length so that it will control the pendulum length during the simulation. You will also need to go to the Model tab in the Model Editor and remove the variable named length from the model table, since this table entry is no longer needed now that the slider control has been created. (To remove the variable length, simple select this entry in the table and click the Remove Selected button just above the table.) Sliders have four parameters that you will want to set. Label this descriptive label appears next to the slider control during the simulation. Value this parameter specifies the initial value, or setting, of the slider. Minimum this parameter specifies the minimum value of the slider. Maximum this parameter specifies the maximum value of the slider. To create a slider control for the initial pendulum angle, click and drag a second slidercontrol from the Component Libraries into your collection of pendulum components and rename it as initialangle. You will also need to go to the Model tab in the Model Editor and modify the Initial Value of the variable theta. Replace the value of 0.5 radians that you entered earlier with either or degreestoradians(initialangle) initialangle With the first version the slider will specify the initial angle in degrees and with the second version the slider will specify the initial angle in radians. The simulation will now reference the slider control initialangle to determine the starting position of the pendulum. Remember to set the values of the four slider parameters.

13 CHAPTER 12. A SIMPLE PENDULUM Suggestions for Further Exploration Once your pendulum simulation is working, here are a few avenues for exploration. Explore the e ect of changing the initial angle or the pendulum length on the period of oscillation. Unlike the usual textbook example where small angles are assumed, the period will depend on both of these parameters. To determine the period of oscillation, go to the Model tab and ensure that the box in the Rec column is checked for the variable theta. Run the simulation by selecting Simulate, Record, and Run from the Run pulldown menu. Upon quitting the simulation, you will find that the numerical values generated for (or theta) are available in the Recorded Data tab at the bottom of the Model Editor and can be used to determine the period. Add components to create a second pendulum, using the small angle approximation, that runs simultaneously with the first one. Compare its simple harmonic motion to the motion of the actual pendulum. Explore the e ect of changing the initial velocity of the pendulum. To do this, simply change the value specified in the Initial Value column for the variable! (or omega) in the Model tab. Modify the model to model a spherical pendulum. A spherical pendulum is similar to the simple pendulum, but without the constraint of a single plane of oscillation. The pendulum position is therefore determined by two angles: as defined above, and an azimuthal angle. The equations of motion for and are d 2 dt 2 = d 2 dt 2 = 0 g L sin