Objective:
To experimentally verify the formula for the dependence of wave speed on the properties of the medium by putting a stretched string into oscillation
Equipment:
An electric string vibrator, a mass scale, string, a pulley with attachments, two sets of rod and C-clamp, a weight hanger, slotted weights, a meter stick, and a sheet of Cartesian graph paper
Theory:
In a medium with constant physical properties, mechanical waves travel at a certain constant speed. The formula for this constant speed is
V = f λ
where in SI, f is the frequency in Hz and λ is the wavelength in meters. If the properties of the medium change, then the wave speed, V, changes. For a string, the mass and length remain constant to a high degree if the string is put under tension. Putting a string under tension affects the speed of waves in it. The formula for this dependence is
where in SI, F is the tension in the string in N, V is the wave speed in m/s, and μ is the mass per unit length of the string in kg/m.
Procedure:
1. Measure the mass of the string using a mass scale. It is okay to use a string that is much longer than necessary. Do not cut the string yet.
2. Measure the length of the string, and calculate its mass per unit length, μ, in kg/m.
3. Mount a rod and C-clamp on the edge of the table and use a skew clamp to connect the string vibrator to the vertical rod.
4. Mount another rod and C-clamp on the same edge (preferably at the corner of the table) and attach the pulley system to it.
5. Attach one end of the string to the string vibrator. Pass its other end over the pulley and connect it to the weight hanger. See Fig.1.
6. Adjust the elevations of the string vibrator and the pulley system such that the string is level or parallel to the edge of the table. Also, make sure that both the vibrator and pulley are at their lowest possible position to minimize the torque they exert on the rods. This guarantees a firmer set up. You may have to adjust the horizontal distance between the vibrator and the pulley from some 70 cm to over 150 cm for more favorable results.
7. Plug in the string vibrator to the electric socket and turn it on.
8. Adjust the amount of weight suspended from the string until you obtain a good standing-wave pattern. Try to develop patterns ranging from one to eight loops if possible. The meaning of the word "loop" as used here needs to be clarified. A loop is a section of string between nodes, but the point at which the string is attached to the vibrator is not necessarily a node. Therefore, for the example shown in Fig. 1, count three loops, not four. Sections AB, BC, and CD form the three loops. The section from point D to the vibrator is not counted.
9. In the table shown below, record the value of the suspended mass M and the effective length L of the string for each case. (The "effective length" is the distance from the top of the pulley, where the string has a node, to the last "true" node near the vibrator. For example, the effective length for the case shown in Fig. 1 is the distance from point A to point D).
10. Complete the calculations for the other columns in the table under Data.
11. If your measurements are correct, the measured value of frequency in the last column should be close to 120Hz. This should be the case for each trial or each line of the table.
12. As shown under "Theory", V = f λ , and since V = (F/μ)^{0.5} ; therefore,
f λ = (F/μ)^{0.5} . Solving for λ results in
This equation has the form y = mx + b , the equation of a straight line, where y corresponds to λ, and x corresponds to . The slope is therefore given by
Plot the wavelength λ versus and fit a straight line to the data. Find the slope of the graph, and calculate the frequency f, using the above equation. This is the measured value for frequency. Compare it with the accepted value of 2 x 60.0 Hz = 120. Hz that is the frequency of oscillations of the string vibrator, and calculate the relevant percent error. The frequency of the vibrator is 120Hz for the following reason. The frequency of the city electric outlet is 60.0Hz, and in each half-cycle the coil (electromagnet) of the vibrator attracts and releases the blade. In each cycle, the blade is attracted and released twice. Therefore, the blade oscillates 120 times per second.
Suspended Mass
(kg) |
Tension in the String F = Mg (N) |
# of
Loops number |
Effective Length (L) (m) |
Wavelength λ =2 (L/n)
(m) |
(N)^{0.5} |
v =
(F/μ)^{0.5} (m/s) |
f = v/λ Hz |
1 | |||||||
2 | |||||||
3 | |||||||
4 | |||||||
5 | |||||||
6 | |||||||
7 | |||||||
8 |
Data:
Given:
f_{accptd.} = 120Hz.
Measured:
The measured values are recorded in the table.
Calculations:
Calculations are performed in the table.
Comparison of the Results:
Conclusion:
To be explained by students.
Discussion:
To be explained by students.