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In this lab you will study standing waves on a string.
First you will measure the frequency of the normal modes at a constant tension.
Next you will measure the frequency of a single harmonic at different tensions.
In each case you can use these measurements to calculate the mass per unit length of the string.
For traveling waves on a string the velocity v
is equal to the square root of the tension over the mass per unit length of the string.
We can also express the velocity as the product at the frequency and the wavelength.
The wavelength is the linear distance for the wave to repeat,
and the frequency is the number of times the wave repeats per unit time.
For transverse waves the wave propagates in a direction perpendicular to the
motion of the particles in the medium.
Standing waves are created at resonant frequencies.
At these frequencies the incoming and reflected wave add constructively to create a standing wave.
Different resonant frequencies are called harmonics.
We label each harmonic with a number n.
The stationary points on the standing wave are called nodes, here marked in red.
The points of maximum amplitudes are called anti-nodes.
The harmonic number n is equal to the number of anti-nodes.
Combining our earlier equations for the velocity of waves on a string,
we find that the frequency is equal to 1 over the wavelength times the square root of the tension over the mass per unit length.
For each harmonic the wave length is equal to two times the length of the string divided by n.
Combining the above equations,
we find an expression for the frequency of the nth harmonic
in terms of the length of the string, the tension, and the mass per unit length of the string.
To start the experiment first measure the mass per unit length of the string
using a triple beam balance and the meter stick.
Tie the string to the blue metal piece on speaker and thread it through the white plastic piece in the center.
Connect the red and black cables to the speaker,
then connect these cables to the power amplifier
and connect the amplifier to the Pasco interface.
Next clamp the metal bridge to the table with the pully
Stretch the string across the table over the metal bridge and hang weights from the string..
You can change the tension on the string by changing the amount of mass.
To control the power amplifier with Data Studio select Channel A,
and then choose power amplifier. A dialog box called signal generator will open.
Here you can select the wave shape, the frequency, and the amplitude.
Choose sine wave for the wave shape.
When adjusting the amplitude do not that it above 5 volts.
You can change the frequency and amplitude while their power amplifier is on by pressing the plus and minus buttons.
You can also set the step size of these changes using a left and right arrows.
Starting at 0 Hz, sweep through a range of frequencies to find a harmonic.
These frequencies will have the largest amplitudes.
This is the first harmonic, it has one anti-node in the center of the string, and two nodes at the ends.
This is the second harmonic.
The node in the center the strain is stationary
You can touch the node without disturbing the standing wave.
If you touch any other place on the string that is not stationary the wave will collapse.
Recording the frequencies for at least five modes
Remembering our equation for the frequency of these modes,
how do you expect the modes to relate to each other?
Then choose a single-mode and find this matter with five different tensions. Record the frequency for each tension.
Once you have your measurements make two plots,
one of the harmonic number versus frequency and the other
of the hanging mass vs frequency for a set harmonic.
From the above equation, what form do you expect for a function and frequency vs mass?
Is there a function of mass you can plot that will give you a straight line?
Once you have straight lines for both plots,
find the slopes and use these slopes to find the mass per unit length of the string.
Does this calculation match your earlier measurement?