1 Harmonic Oscillations and Waves
In addition to reading this assignment, also read Appendix
A on uncertainties and Appendix B on linear
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.
The force exerted by a spring displaced from its equilibrium length is
given by Hooke's law
where 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
The prediction of the frequency of the mass-spring system, derived in
class with the assumption of a massless spring, is
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
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.
The first three standing-wave modes of a string fixed at both
ends and held under tension.
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,
Each value of corresponds to a different standing-wave
mode of the string. The standing-wave mode corresponding to
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
The speed of waves on a string is given by
where is the tension in the string and 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.
- Make sure the force probe and motion detector are connected to
the Labpro interface.
- Remove the spring from the force probe.
- Make sure the switch on the force probe is in the 50 N
- Run Logger Pro, and open the file
- 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 () of the hanging mass.
- 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.
- Use the balance at the front of the lab to measure the mass of the
spring, and record the result and its uncertainty.
- Hang the spring from the hook on the force probe. Orient the
spring with its narrow end up.
- Hang 200 g from the spring (hanger + 150 g).
- Place the motion detector on the floor directly below the
hanging mass, so that it is ``looking'' up.
- 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
) 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
- Collect a clean 20-second-long measurement.
- Run the string over the pulley, and hang a mass hanger from the
- 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.
- 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 ...
- Record the value of for the mode.
- 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.
- Measure the length of the fixed ends of the
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.
- 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.
- 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
- 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.
- Calculate the frequency predicted assuming a massless spring and
- Then, use your measured frequency to calculate the effective
mass of the spring
and its uncertainty.
- Put your raw data (, , ) into a spreadsheet, and
calculate the linear density of the stretched string and the
associated uncertainty for each of the
standing-wave modes you observed.
- Devise a strategy for determining, based on your results,
whether or not the linear density of the string depends
significantly on tension.
- Devise a method of determining a best value of the linear
density of the string and its uncertainty from your four results.
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.
- Describe your strategy for extracting the frequency
of the mass-spring system and its uncertainty from your Distance
vs. Time graph. Report your result.
- What percentage of the actual mass of the spring is
? Give reasoning for the hypothesis that
must be less than the mass of the spring.
- 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.
- 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.
- Describe the method you used to determine a best value of the
linear density of the string and its uncertainty. Report your