1. Preliminary Questions

Oscillations

There are many forms of oscillations. A string on a guitar, a vibrating tuning fork, and a monkey swinging on a vine are examples. Regardless of whether an oscillation is simple or complex, the motion can be described using a sine function or a combination of sine functions. Moreover, since sine functions are related to the components of position of a point moving in a circular path, there are close connections between circular motion and oscillations. In this activity, you will explore these connections using a video of three oscillating systems: a pendulum, a mass on a spring, and a dot on a rotating disk.

1. Preliminary Questions

Note: You will receive full credit for each prediction made in this preliminary section whether or not it matches conclusions you reach in the next section. As part of the learning process it is important to compare your predictions with your results. Do not change your predictions!

(a) The simplest form of oscillatory motion can be described by a sine function, which we can write in the general form . Suppose y is the vertical position of a mass hanging from a spring, measured in meters, and t is time, measured in seconds. What are the units of the constants A, B, C and D? Assume that the sine function is defined for an angle in radians.

(b) Sketch graphs of one cycle of y = sin(t) and y = cos(t) on the axis. Don’t worry about units or numbers, just get the shapes right. In particular, think about the relative value of each function at the origin, t = 0. (i.e., are the values at a minimum, zero or maximum for each function?) Use this sketch as a guide later in the activity, when you have to answer questions such as, “Does the graph look like sin(t)?” Label each curve.

(c) When the graph of the sine function sin(t) is shifted to the left by 90 degrees or π/2 radians, it looks exactly like the graph of the cosine function cos(t). Circle the equation that correctly represents this fact:

cos(t) = sin((π /2)t) cos(t) = sin(t – π /2)

cos(t) = sin((–π /2)t) cos(t) = sin(t) + π /2

cos(t) = sin(t + π /2) cos(t) = sin(t) – π /2

2. Activity-Based Questions

Do the components of circular motion behave as sinusoidal oscillations? Start your investigation by opening the Logger Pro Experiment file <Oscillations-1.cmbl>. It has the movie <ThreeSystems-1.mov> inserted in it. This movie shows the dot on the disk moving at a constant speed while the pendulum and mass on a spring are resting. Ignore the VA_Angle window for the time being. The movie has already been calibrated in meters.

(a)  Grab the lower left corner of the movie window and drag it to make the movie larger. Click the AddPoint tool () and carefully mark the center of the blue/yellow spot on the disk in every frame. In the Page menu choose Auto Arrange to make the graph visible again. It shows the y component of the dot’s position versus time. Does this graph look like the graph of sin(t)? How is it the same, and how is it different?

(b)  Select the graph by clicking on it once. In the Analyze menu choose Curve Fit. From the list of possible functions, choose the sine function and then Try Fit and OK. Is it a good fit? Explain why or why not. What are the values of the fit parameters A, B, C and D, with units?

(c)  Sketch the graph of the fit on the axes shown in Figure 1. Draw it as a solid line.

(d)  Click on the Show Origin tool (). This reveals the coordinate axes. Note that the origin has been placed at the center of the disk. The parameter VA_Angle shown in a window below the movie is the tilt angle of the x-axis. In the window with the VA_Angle display, click on the Increase Angle tool () about 10 or 15 times to tilt the axes. Describe what the graph appears to do while you change the angle. Which fit parameters stay about the same and which change? Which parameter changes the most, and does it get bigger or smaller?

(e)  Adjust the angle until the fit line starts at the origin (as a graph of sin(t) would). If you double-click on the window showing the value of VA_Angle, you’ll get a dialog box that allows you to set the angle. Watch what happens to the fit parameter that changes the most. When you tilt the axis a little more than or a little less than the angle where the fit line looks like sin(t), this parameter should switch between zero and 2π = 6.283. Set the parameter as close to zero as you can get it by carefully adjusting VA_Angle. What is the value of VA_Angle now? When you convert it to radians, how does it compare with the parameters you found in part 2(b)?

(f)  Sketch the graph of the fit on the axes shown in Figure 1 on the previous page. Use a dotted line.

(g)  Parameter C is called the “phase angle” of the sine curve. It is often represented by the Greek letter . As you have seen, for the x-coordinate of a point on a circle, the phase angle tells how much you have to rotate the axes to make the graph of x(t) look like a sine function sin(t). Rewind the movie (). What is the relationship between the dot on the circle and the rotated x-axis in the initial frame?

(h)  Double-click on the graph to get the Graph Options dialog. Choose Axes Options, set it to display both X_circle and Y_circle on the graph and click OK. Fit X_circle to a sine function. Does the graph of the fit look like sin(t)? If not, what function does it look like?

(i)  You have already set the axes so the phase angle for Y_circle is zero. Convert the phase angle for X_circle from radians to degrees. Is this the value you expect? Why or why not?

(j)  Parameter A is called the “amplitude” of the function. Do X_circle and Y_circle have about the same value of A? Should they be the same? Explain why or why not.

(k)  Parameter D is the “offset” of the function. It tells how far the center of the oscillation is from the coordinate origin. If the origin is at the center of the circle, D should be zero. Is it close to zero for X_circle and Y_circle?

(l)  Parameter B is called the “angular frequency” of the function. It is often represented by the Greek letter w. To understand what this means, first determine the time it takes for the dot to go around the circle once. On the graph, this will be the time interval from a point where the graph crosses the time axis to the next time where it crosses with the same slope. Select the graph. With the mouse, point to the third place where the Y_circle graph crosses the time axis. (You adjusted the phase angle so the first place it crosses is the origin of the graph.) The coordinates of that point are shown in the lower left corner of the graph window. What is this value (with units) of the time interval to three significant digits? It is called the “period” of the motion and is denoted T.

(m) Calculate 1/T. This is the “frequency” f of the motion. Write its value with units.

(n)  Multiply the frequency f by 2π radians and write the result here to three significant digits with appropriate units. Is the result nearly equal to the value you obtained for the angular frequency? Express the relationship between w and f as an equation.

How can you compare the phases of different systems? Start this investigation by opening the LoggerPro Experiment file <Oscillations-2.cmbl>. It has the movie <ThreeSystems-2.mov> inserted in it. The pendulum length, the mass and the motor speed were all adjusted so the systems would have about the same period. The positions of the dot on the circle, the pendulum bob and the mass on the spring have already been marked for you. To see the points, step through the movie using the Next frame button () under the movie.

(o)  Do X_circle and X_pend appear to have about the same phase? Explain how you can tell by looking at the graph or the movie.

(p)  Fit both X_circle and X_pend to sine functions as you did earlier. What are their phase angles? Are they about the same?

(q)  Convert the phase angle of X_circle to degrees and write the result here. Double-click on the VA_Angle window to open a dialog. Where it says Value, type in the phase angle of X_circle in degrees, and click OK. Does the X_circle graph look like sin(t) now? How about the graph of X_pend? Explain what happened to it. Why can’t you change its phase by rotating the axes?

(r)  Set VA_Angle back to zero. Double-click on the graph to get the Graph Options dialog. Choose Axes Options, set it to display both X_circle and Y_mass (but not X_pend) on the graph and click OK. Fit Y_mass to a sine function. Which parameters are about the same for the two systems? Which are quite different? Which are nearly zero?

(s)  As you saw above for X_pend, you cannot change the phase of Y_mass by rotating the axes. However, you can compare the phases using the fit parameters. Find the phase difference by subtracting the phase angle of X_circle from the phase angle of Y_mass, and convert the result to degrees. Show your calculation. Explain how the positive or negative sign of the phase difference tells you which graph is a little to the right or left of the other.

(t)  Rewind the movie () and step back and forth through the first five frames using the Previous frame () and Next frame () buttons. Notice what the dot and mass are doing. Explain how this visual information confirms the phase difference you calculated above for question (s).

(u)  Earlier when you compared the fits for X_circle and Y_circle, you found that they had the same amplitude and angular frequency. Do X_circle and Y_mass have the same amplitude and angular frequency? Should they? Explain why or why not.

Can you recognize properties of oscillatory motion without making graphs? Open the Logger Pro Experiment file <Oscillations-3.cmbl>. It has the movie <ThreeSystems-3.mov> inserted in it. The movie shows the same systems, but the oscillations are a bit different than those in the ThreeSystems-2.mov. Click the Play button () under the video to see how they move now. By looking carefully at the movie frame-by-frame, and not making any graphs, answer the following questions.

(v)  Which of the systems has the longest period? Which has the shortest period? How can you tell?

(w)  Which of the systems has the largest amplitude? Which has the smallest? How can you tell?

3. Reflections on Your Findings

(a)  In 2(n) you wrote the value of angular frequency with units. Do those units agree with your answer to preliminary question 1(a)? If not, explain the difference.

(b)  In 2(s) you explained how the sign of a phase difference tells which graph is on the right of another graph. Use the same reasoning to decide whether or not cos(t) equals sin(t – π/2).

Physics with Video Analysis 18 - XXX