DescriptionStanding Waves on a String (Online)
The purpose of this experiment is to generate a transverse wave on a string and measure the tension that causes a
standing wave. Using the tension to observe the relationship between tension and the wavelength. Finally to confirm
the wave relationship = holds.
Online Simulation https://ophysics.com/w8.html
Excel, Google Sheets, or other spreadsheet program.
With a stretched string, the wave the travels along the string is defined by the tension and linear mass density. The
equation can be written as
string = √
If one end of the string is held fixed and the other end is attached to a vibrator that oscillates in the vertical direction.
These waves will be reflected off the fixed end and will be inverted such that the two waves will then interfere. If the
tension is adjusted just right a standing wave will be produced. The figure below shows the following wave patterns with
increasing numbers of “loops.” These are points where the string has maximum displacements.
1. Open the simulation. Set the frequency to as close to 120 Hz as possible. Record your frequency value in Table 1
in the actual column. Set the linear mass density to something around 0.42 × 10−3 kg/m. Record the length
between the oscillator and the pulley. Calculate the wavelength, , for each number of loops and record in Table
2. Using the Tension slider change the tension in the string until you get 5 loops.
3. In Table 1 record the tension. Use the equation for the speed of the wave on a string = √ to solve for the
velocity of the wave. Finally use the fundamental wave property = to solve for the frequency. Record in
4. Using the frequency in step 3 compare that to the actual frequency set by the simulation. Use the percent error
calculation. Record your value in Table 1.
5. Repeat this step for “loops” 4, 3, and 2.
= √ (m/s)
6. Using a spreadsheet program plot the square root of the tension as a function of the wavelength. This means
your y-axis should be the √ and your x-axis should be .
7. Add a trendline to your graph. Add your graph with the trendline in the space below.
8. The slope of the trendline should be equal to the product of the frequency and the square root of the linear
density. Record the slope below and your calculation for √ . Record your observations on if the two are close
9. Make a prediction on the tension force required to get a single loop (n=1). Show all your work below. You can
insert a picture of your work.
10. Since the force is too large for the simulation, suggest a way to modify your setup in the simulation to allow a
single loop standing wave. Record your suggestion below.
11. Using the new modified setup calculate the theoretical tension required. Record your calculations below. Linear
12. In the simulation adjust the parameter you suggested in step 9 and find the tension required for a single loop.
Record this actual tension.
13. Calculate the percent error between the tension you found in step 11 and that you found in step 12. Record
your answer below.
The Simple Pendulum
Dr. Jonathan Meair
A simple pendulum takes the same amount of time to complete
each cycle of its motion. When considering what could affect
the time per cycle or period of the motion, we can consider at
least three factors:
The amplitude of the swing
The mass of the pendulum bob
The length of the pendulum
To determine how these factors affect the pendulum period,
you can do a controlled experiment in which the period of
oscillation is measured while only changing one variable at a
time. In this experiment, you will use a stopwatch to measure
the period of a pendulum and determine how it is affected by
each of the described factors.
Students will determine how the period of a simple
pendulum depends on its amplitude, mass, and length.
You will need string and some objects to hang from it. For string you can use dental floss, thread
from a sewing kit, fishing line, shoe laces, etc. Be creative, but be sure that the mass you hang
from it is much heavier than the string.
To make measurements you will need a stopwatch (such as the one on your cell phone) and a
ruler or some other precise measurement tool. If you cannot find a ruler, remember that US letter
paper is 8.5 X 11.0 inches and 1 inch = 2.54 cm. You should be able to do a fine job making
length measurements if you put in the effort.
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Cut a string that is at least 1 meter long and tie it to something stable so that it hangs
vertically. Hang an object from the free end of the string. Time how long it takes for 10
cycles to occur. Record the value:
t = _________
What is the period of the oscillation? Show your work.
T = _____________
Repeat the experiment several times, but releasing the mass at different initial angles (but
smaller than 30 degrees.) Record the periods below.
T1 = ____________
T2 = _____________
T3 = ____________ T4 = ____________
Did the angular amplitude of the oscillation have a significant effect on the period of
oscillation? Is this consistent with the theoretical model for the period of oscillation of a
pendulum? Carefully explain your answer using well-written complete sentences.
2 of 6
Repeat experiment 1, except this time use a consistent angular amplitude but change the
object that you hang with objects of different weights. Be careful to make sure that the
length of the pendulum as measured from the point of support to the center of mass of the
hanging object does not change from one trial to the next. Record the period for each of
the trials below.
T1 = ____________
T2 = _____________
T3 = ____________ T4 = ____________
Did the mass of the hanging object have a significant effect on the period of oscillation?
Is this consistent with the theoretical model for the period of oscillation of a pendulum?
Carefully explain your answer using well-written complete sentences.
3 of 6
Choose one mass and one angular
amplitude to work with. Record the
period of oscillation (based on 10
complete oscillations) for 6 different
pendulum lengths between 0.25
meters and 2 meters. Create a table
with two columns in which to record
your data. The first column should
be the pendulum length L and the
second should be the period T.
Record your data.
Carefully create a scatter plot of the period squared (T2) versus the pendulum length (L).
Carefully label axes and add a trend-line. Be sure to display the equation on the chart. See
the instructional video on graphing with Microsoft Excel. Also see the “Data Analysis
and Graphing” instruction file for information about graph formatting. Include a
screenshot of your graph here or attach a scanned copy of your graph to the back of this
4 of 6
9. Use the model for how the period of a pendulum squared depends on its length along
with the fit parameters of your linear fit to determine an experimental value for the freefall gravitation acceleration on Earth. You may not need both fit parameters. Show your
work. (You may find it helpful to review the “Data Analysis and Graphing” instruction
file to assist in understanding how to compare models to fit equations.)
10. What is the percent error in your experimental value for the free-fall gravitational
acceleration (g) of Earth? Show your work.
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11. Conclusion Question: According to the model, how does the period of a pendulum
swing depend on its length? Carefully state the mathematical relationship between them.
Does your experiment confirm this relationship? Justify your answer by referencing
important features in your graph. Carefully explain your answer using well-written
12. Take a picture of your experimental setup (including the objects used) and put it in the
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