Write your name, section, and today's date in the cell below:¶
Name - Section - Date -
Introduction¶
A string attached to a turning fork is set vibrating at the same frequency as the tuning fork. The length and tension in the string are adjusted until standing waves are observed on the string. By knowing the tension in the string and the wavelength of the standing waves, the frequency of oscillation of the string and thus, the tuning fork is found.
Theory¶
If transverse waves of constant frequency and amplitude are sent down a string that is fixed at one end, then a reflection of the waves occurs and the oppositely directed waves interfere with each other to produce standing waves. A wave traveling in the positive x direction is given by $$ y_+ = A \sin \left(kx -\omega t \right) $$ where A is the amplitude, k = 2\pi / \lambda is the wavenumber with \lambda being the wavelength, and \omega= 2\pi f with f being the frequency. For a similar wave traveling in the negative x direction with the same amplitude $$ y_- = A \sin \left(kx +\omega t \right) $$ If the amplitudes of the waves are small, then the waves obey the law of superposition and add linearly. The resultant wave is $$ y = y_+ + y_- = 2 A \cos \omega t \sin kx $$ which represents a standing wave whose amplitude, 2A \cos \omega t is a function of time. Figure 2 shows the standing wave. The diagram indicates that the wave shape is not moving along the string but is only oscillating vertically on the string.
Figure 1: Standing wave.
The frequency of a wave is given by $$ f = \frac{v}{\lambda} \, ,$$ where v is the speed at which the transverse waves propagate along the string, The speed of the wave, in terms of the tension T, and the mass per unit length of the string \mu is $$ v = \sqrt{T/\mu} \, . $$ The frequency is, therefore, $$ f = \frac{v}{\lambda} = \frac{1}{\lambda}\sqrt{T/\mu} \, .$$
Setup¶
This experiment consists of a tuning fork mounted to the lab bench, which vibrates a horizontal string that can be draped over a pulley and loaded with mass as in Figure 2 below.
Figure 2: Schematic transverse wave experimental setup.
Adding mass to the hanger will change the tension in the string. This will change the wave speed in the string. With the frequency and string length fixed throughout the experiment, we will generate different standing wave modes with different wavelengths. We will measure the wavelengths of the various standing waves and find the relationship between tension and wavelength.
Procedure¶
1. Attach the mass hanger to the free end of the string over the pulley¶
Use the C-clamp to secure the tuning fork to one end of your lab table. Make sure that the string is as long as possible and parallel to the lab table. Adjust the spark gap on your tuning fork for efficient operation (ask for help if necessary). Measure the length of your string from the pulley to the tuning fork and record it in the cell below.
Record the length of string here (include units!)¶
L =
2. Add some mass to the hanger and turn on the tuning fork vibrator¶
Do NOT supply more than 7 volts DC to the vibrator! You probably will not see a standing mode initially. Now it is time to adjust the weight on the hanger.
3. Observe six or seven normal modes of vibration¶
Producing the fundamental oscillation mode requires about 1.5 kg on the hook. Find the fundamental and record the mass required in the cell below. The higher normal modes are obtained by decreasing the mass on the hook. Slowly change the amount of mass on the hook. Try and achieve the best precision possible when you measure the mass on the hanger. What is the error for your measurements? Eventually, the mass of the empty hook (50 g) will limit the number of normal modes that you can observe. Try to find as many as possible so that your mean measurement has as small an error as possible (the “Error on the Mean”).
Record the mass required to achieve fundamental oscillation mode and the wavelength here¶
Fundamental (n=1) mode m = λ =
Uncertainty of measurements: Δm = Δλ =
4. Normal Modes¶
- Determine the tension in the string, T = mg where m is the mass in kilograms, and g = 980 cm/s^2 is the average acceleration due to gravity at Earth’s surface.
- Determine the wavelength of oscillation for each normal mode by dividing the length of the string in centimeters by the number of anti-nodes in each mode.
- Determine the normal mode, n = (N − 1), where N is the number of nodes. Remember: There is a node at each end where the string is fixed.
- Repeat and find as many different normal modes as possible. Record all data in the cell below.
Record the mass required to achieve normal oscillation modes and the wavelengths here¶
n=2 mode m =
n=3 mode m =
...
Uncertainty of measurements: Δm = Δλ =
6. Find the Linear mass density (\mu) of the string¶
Use the long string provided by the instructor to determine the linear mass density of the string and record your measurements in the cell below.
Record the length, mass, and linear mass density of the string sample here¶
L = m = µ =
Uncertainty of measurements: Δm = ΔL =
Calculate the uncertainty in the linear mass density using the expression $$ \frac{\Delta \mu}{\mu} = \sqrt{ \left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta L}{L}\right)^2 } \, . $$
Uncertainty of linear mass density: Δµ =
Why is it preferable to use a large length of string for measurement of the linear mass density?
Provide your response here¶
Analysis¶
- Plot the m vs. \lambda^2 for the observed normal modes, where m is mass, and \lambda is wavelength. Put \lambda^2 on the horizontal axis, and m on the vertical axis. Don’t forget to include an appropriate title on your plot and indicate units on the axes.
## Create your plot here by modifying the dummy code below
## Be sure to label your axes and include units
## Remember you can find matplotlib documentation at https://matplotlib.org/stable/users/index.html
import matplotlib.pyplot as plt
import numpy as np
wavelengths = [ 1., 2., 3.]
masses = [3., 2., 1]
squared_wavelengths = np.array(wavelengths)**2
fig, ax = plt.subplots()
ax.plot(squared_wavelengths, masses, label = 'test')
ax.set(xlabel='x-axis', ylabel='y-axis', title='Sample plot for demonstration')
ax.legend()
ax.grid()
#fig.savefig("test.png") #This allows you to save the plot to a file
plt.show()
Is your plot consistent with a straight line? Why or why not?
Provide your response here.¶
- Starting from the formula relating linear frequency, string tension, linear mass density, and wavelength, $$ f = \frac{1}{\lambda}\sqrt{T/\mu} $$ Write a formula for the linear frequency of vibration in terms of measured quantities only (hanger mass, length, string mass, and number of nodes).
Write your expression here¶
- Write an expression for the uncertainty in frequency for the measurements of a single normal mode. The general expression for the uncertainty of a quantity A(x,y), assuming that errors in x and y are uncorrelated is given by $$ \Delta A(x,y) = \sqrt{ \left( \frac{ \partial A }{ \partial x } \right)^2 \Delta x^2 + \left( \frac{ \partial A }{ \partial y } \right)^2 \Delta y^2 } \, . $$
Write your expression here¶
- Combining k independent measurements of the same quantity A, each with uncertainty \delta A leads to an overall uncertainty $$ \Delta A = \frac{\delta A }{\sqrt{k}} \, . $$ Using your measurements above, provide an estimate of the frequency of the tuning fork and an overall uncertainty for your inference.
Provide your response here, writing out how you arrived at your answer.¶
Best estimate: f = Uncertainty: Δf =
- Is the manufacturer’s linear frequency f = 80 Hz within your error bounds? In other words, can you verify that the tuning fork is operating as advertised within the uncertainty on your measurement?
Provide your response here.¶
- Identify at least two sources of statistical error and comment on how to reduce these errors.
Provide your response here.¶
- Identify at least two sources of systematic error and comment on how to reduce these errors.