Subsections


1 Harmonic Oscillations and Waves

In addition to reading this assignment, also read Appendix A on uncertainties and Appendix B on linear regressions.

Introduction

In order to speed the checkout procedure at the end of lab, please discuss your data and preliminary answers to the questions in section 2 with your instructor as the lab progresses.

A Mass/Spring System

The force exerted by a spring displaced from its equilibrium length is given by Hooke's law

\begin{displaymath}
F^\mathrm{spring} = -k x
\end{displaymath} (1)

where $x$ is the displacement of the spring measured relative to its equilibrium position. You will study the motion of a mass hanging vertically from a spring. The only effect gravity has on a spring-mass system hung vertically is to shift the equilibrium position of the system.

The prediction of the frequency of the mass-spring system, derived in class with the assumption of a massless spring, is

\begin{displaymath}
\omega = 2 \pi f = \sqrt{\frac{k}{m}}
\end{displaymath} (2)

However, since you are working with a real spring - one with mass - this prediction will fail. We can correct for the mass of the spring by including an effective mass in the prediction
\begin{displaymath}
\omega = 2 \pi f = \sqrt{\frac{k}{m + m^\mathrm{eff}}}
\end{displaymath} (3)

You will find empirically that the effective mass is less than that of the entire spring, and you will be asked to think about why that makes sense.

Standing Waves on a String

Figure 1: The first three standing-wave modes of a string fixed at both ends and held under tension.
\scalebox{0.8}{
\includegraphics{HOandW-stand.eps}
}

A string fixed at both ends and held under tension can display modes of oscillation called standing waves in which waves traveling back and forth between the fixed ends interfere constructively. Standing waves only occur at frequencies at which an integer number of half-wavelengths fit between the fixed ends of the string,

\begin{displaymath}
L = \frac{\lambda}{2}   n \hspace{1cm} (n=1, 2, 3 \ldots)
\end{displaymath} (4)

Each value of $n$ corresponds to a different standing-wave mode of the string. The standing-wave mode corresponding to $n=1$ is called the fundamental mode. The first three standing-wave modes are shown in Figure 1. The frequencies of the standing-wave modes are given by
\begin{displaymath}
f = \frac{v}{\lambda} = \frac{v}{2L}   n \hspace{1cm} (n=1, 2, 3 \ldots)
\end{displaymath} (5)

The speed of waves on a string is given by
\begin{displaymath}
v = \sqrt{\frac{T}{\mu}}
\end{displaymath} (6)

where $T$ is the tension in the string and $\mu$ is the linear density of the string, measured in units of mass per length.

You will study the behavior of a string fixed at both ends and driven by a harmonic oscillator. You will vary the tension in the string to find and observe several standing-wave modes of the system and test the theoretical predictions above. You will also determine the linear density of the string.

Experiments

A Mass/Spring System

  1. Make sure the force probe and motion detector are connected to the Labpro interface.

  2. Remove the spring from the force probe.

  3. Make sure the switch on the force probe is in the 50 N position.

  4. Run Logger Pro, and open the file SHO.cmbl.

  5. Check the calibration of the force probe by hanging a known mass in the 100-500 g range from it. Collect force data for a few seconds and use Analyze -> Statistics to find the average force. Compare this with the weight ($mg$) of the hanging mass.

  6. If the calibration is off, calibrate the probe. Click on Experiment -> Calibrate -> LabPro: 1 CH1: Dual Range Force and select the Calibrate tab. Then hang 50 g from the sensor (an empty mass hanger), enter the force (0.49 N) and click on Keep. Repeat the process with 550 g (5.39 N). Then, check the calibration again.

  7. Use the balance at the front of the lab to measure the mass of the spring, and record the result and its uncertainty.

  8. Hang the spring from the hook on the force probe. Orient the spring with its narrow end up.

  9. Hang 200 g from the spring (hanger + 150 g).

  10. Place the motion detector on the floor directly below the hanging mass, so that it is ``looking'' up.

  11. Displace the mass vertically, and release it gently from rest, and click on Collect. The first time you collect data, you may not see anything on the graphs, because the scales of the graphs may not be properly set for your measurements. Click on the autoscale button ( \scalebox{0.7}{\includegraphics{LoggerPro3-autoscale.eps}}) on the left side of the Logger Pro toolbar.
    Caution! Beware of exciting both longitudinal and transverse standing-wave modes of the spring. They will appear as fast wiggles riding on the overall oscillations in the Force vs. Time graph. Practice releasing the system until you can get a nice long measurement without significant ``standing-wave contamination.'' If it's unclear what you're supposed to avoid here, consult with your instructor.

  12. Collect a clean 20-second-long measurement.

Standing Waves on a String

  1. Run the string over the pulley, and hang a mass hanger from the free end.

  2. Plug in the power cord of the blue transducer. The frequency of the transducer is a constant 60.00 Hz. The uncertainty in the frequency is negligible compared with uncertainties in your other measurements.

  3. Vary the amount of mass hanging from the string to find several standing-wave modes of the string. (Do not use more than 750 g.) Carefully adjust the mass to give the largest amplitude.

    For each of four different standing-wave modes ...

    1. Record the value of $n$ for the mode.

    2. Record the mass hanging from the string and its uncertainty. Determine the uncertainty by finding the largest amount of mass you can add without significantly reducing the amplitude of the standing wave.

    3. Measure the length $L$ of the fixed ends of the string.
      Caution! The point at which the string is tied to the transducer is not necessarily a fixed point! In some cases, you may find that there is a node very close to the transducer. You should measure the length from the node rather than from the transducer.

    4. Attempt to measure the percentage by which the string is stretched by the hanging mass. You'll need to devise a strategy. You may want to consult with your instructor.

Analysis

A Mass/Spring System

  1. What is the physical meaning of the slope of the Force vs. Distance graph? You need this quantity. Use Analyze -> Curve Fit ... to find it. Record it and its uncertainty.

  2. Devise a method of extracting the frequency of oscillation of the system and its uncertainty from your Distance vs. Time graph. Discuss your method with your instructor.

  3. Calculate the frequency predicted assuming a massless spring and its uncertainty.

  4. Then, use your measured frequency to calculate the effective mass of the spring $m^\mathrm{eff}$ and its uncertainty.

Standing Waves on a String

  1. Put your raw data ($m$, $n$, $L$) into a spreadsheet, and calculate the linear density $\mu$ of the stretched string and the associated uncertainty $\sigma_\mu$ for each of the standing-wave modes you observed.

  2. Devise a strategy for determining, based on your results, whether or not the linear density of the string depends significantly on tension.

  3. Devise a method of determining a best value of the linear density of the string and its uncertainty from your four results.


Checkout

Be prepared to answer the following questions (and more) during and at the end of the lab period. When you think you are ready to checkout, inform your instructor. Your instructor will pose questions to your group and individuals within your group. The instructor must be satisfied that all members of the group understand the concepts, procedure, and data. If the instructor is satisfied you have finished the lab.

You can make this process more efficient by answering questions throughout the lab period. Call your instructor over to discuss your data and your preliminary thoughts on any of the following questions.

Always give reasoning with your answers.
  1. Describe your strategy for extracting the frequency of the mass-spring system and its uncertainty from your Distance vs. Time graph. Report your result.

  2. What percentage of the actual mass of the spring is $m^\mathrm{eff}$? Give reasoning for the hypothesis that $m^\mathrm{eff}$ must be less than the mass of the spring.

  3. Describe your strategy for finding the percentage by which the string was stretched by the hanging mass in your standing wave measurements. Report your results.

  4. Give all of your linear density results with uncertainties. Describe how you determined, based on your results, whether or not the linear density of the string depends significantly on tension. Give your conclusion.

  5. Describe the method you used to determine a best value of the linear density of the string and its uncertainty. Report your result.