Lab 1: Sonometer
Tsai-Yen (Yoyo) Wang
PHYS-UA-12: General Physics II Lab, Section 003
Joan Manuel La Madrid
Performed February 7th, 2022
Due February 14th, 2022
Objective
The overall objective of this lab was to measure and understand how normal mode
frequencies of metal wires that are fixed at both ends change with the following four parameters:
length, mode number, tension, and mass per unit length of a wire. The lab was composed of four
experiments, in each of which one of the four parameters was altered while the other three
parameters remained constant. In this way, experimenters were able to observe how each of the
four parameters is related to the frequency of a metal wire.
Description
The key equipment used in this lab included: a PASCO sonometer with detector coil,
Capstone software, voltage sensor, a 1 kg hanging mass, and two sets of wires. Similar to a string
on a guitar, a wire is held under tension on a sonometer, a device used to measure the normal
vibrations of the wire.
To adjust the length of the vibrating part of the wire, two bridges could be moved to the
desired position. To change the tension of the wire, a hanging mass was hung onto one of the five
notches on the tensioning lever. Tuning pegs needed to be adjusted so that the tensioning lever
could be horizontal for an accurate tension value. To detect the oscillations of the wire, the
detector coil was placed right underneath the wire. Since the distance from the lever pivot to the
wire was the same as that from pivot to notch 1, the tension created by the lever arm in the wire
was equal to 1Mg, where M represents the mass of the hanging mass. As the hanging mass was
moved to Notch 2, the tension would increase to 2Mg, and so on.
When the wire was plucked, its vibration initiated a slight change in the magnetic field of
the coil, resulting in a small ac voltage across the end of the coil through induction. This signal
from the detector coil was then detected by a voltage sensor. To show the frequencies detected in
the signal, Capstone was set up with a Fast Fourier Transform (FFT) display which allowed the
experiment to monitor the oscillations of the wire and determine the values of normal mode
frequencies.
Figure 1. Device Overview of PASCO sonometer
Theory
Consider that a wire is stretched between the two bridges on a sonometer, just like the
setup for this lab. When the wire is struck, it is made to oscillate in a standing wave pattern (also
known as a normal mode), vibrating in several harmonics at the same time. The lowest frequency
of the standing wave is called the fundamental harmonic, which is observed to have the
maximum amplitude. Subsequent harmonics with higher order modes are integer multiples of the
fundamental harmonic.
When the wire is fixed at both ends separated by a fixed distance L, wavelengths of
resonant frequencies or normal mode wavelengths can be expressed as λ =
2𝐿
𝑛
(𝑛 = 1, 2, 3...),
where n is an integer representing the mode number. The lowest or fundamental mode would
have n = 1, so the normal mode wavelength would be twice the wire length (λ = 2𝐿) in this
case. In addition, the wave speed on a stretched wire is set by the tension T and mass per unit of
the wire µ, which is given by the following equation 𝑣 =
𝑇
µ
.
The wave speed v is related to the wavelength and the frequency by 𝑣 = 𝑓λ , so one can
express normal mode frequencies as 𝑓 =
λ=
2𝐿
𝑛
𝑇
µ
and 𝑣 =
rewritten as 𝑓𝑛 =
𝑣
λ
. By replacing the wavelength and wave speed with
, the normal mode frequencies for a wire that is fixed at both ends can be
𝑛
2𝐿
𝑇
µ
. This is the equation that would be explored in the experiments. As
shown in the equation, normal mode frequencies are related to four parameters: length (L), mode
number (n), tension (T), and mass per unit length (µ) of a wire. Thus, using a sonometer would
allow the experimenter to vary one parameter while keeping the other three constant. In this way,
one would be able to investigate how normal mode frequencies are affected by each parameter.
Data and Calculations
5.1 Exploring Normal Modes
Theoretical value for fundamental frequency
𝑓1(𝐻𝑧)
=
𝑛
2𝐿
𝑇
µ
=
1
2(0.5𝑚)
3*1𝑘𝑔*9.81
−3
1.84×10 𝑘𝑔/𝑚
= 126. 470𝐻𝑧 ≈ 126 𝐻𝑧 (3 𝑠. 𝑓.)
Table 1. Mode frequencies versus mode number
n
𝑓𝑛(𝐻𝑧)
𝑓𝑛
𝑛
(𝐻𝑧)
1
131.065
131.065
2
264.700
132.350
3
403.475
134.492
4
537.109
134.277
Average
133.046
Figure 2. Graph of the oscillations displaying normal mode frequencies from 𝑓1 to 𝑓4 . This
graph illustrates the 4 normal mode frequencies that the experimenters were able to determine
using a wire fixed at both ends separated by a distance of 0.50 m and held under a tension of 1.
The fundamental frequency is 131.065 Hz and displays the strongest signal, shown by the
greatest value of voltage. The mode number and normal mode frequency are linearly related
when the length (L), tension (T), and mass per unit length (µ) of a wire are kept constant. Since
the mode number (n) is a positive integer (1, 2, 3…), the values of normal mode frequency
appear to be the multiple of the fundamental frequency, which appears to be 131.065 Hz in the
graph.
5.2 Explore Changing Length
Table 2. Mode frequencies versus L
𝐿(𝑚) = 𝑋2 − 𝑋1
𝑓𝑛(𝐻𝑧)
0.40
161.904
161.904 × 0.40 = 64.762 ≈ 65
(2 s.f.)
0.50
131.065
65.532 ≈ 66 (2 s.f.)
0.60
108.214
64.928 ≈ 65 (2 s.f.)
0.65
97.656
63.476 ≈ 63 (2 s.f.)
𝑓𝑛𝐿(𝑚/𝑠)
5.3 Exploring Tension
Table 3. Mode frequencies versus Tension
Notches
T (N)
𝑓𝑛(𝐻𝑧)
𝑓1
1kg*9.81 𝑚2 = 9.81 N
71.263
71.263
2
19.6
102.935
23.251 ≈ 23. 3 (3 s.f.)
3
29.4
129.329
23.852 ≈ 23. 9 (3 s.f.)
4
39.2
150.443
24.029 ≈ 24. 0 (3 s.f.)
5
49.0
166.280
23.754 ≈ 23. 8 (3 s.f.)
1
𝑇
𝑠
𝐻𝑧
(
𝑁
)
=22.764 ≈ 22. 8 (3 s.f.)
9.8
Table 4. Mode frequencies and Calculating Mass Density (µ)
n
𝑓𝑛(𝐻𝑧)
1
129.329
µ(
𝑘𝑔
𝑚
)
2
µ=
2
𝑇𝑛
2
2
𝑓𝑛 (2𝐿)
2
256.018
0.00180
3
387.986
0.00176
4
519.964
0.00174
=
(3×1𝑘𝑔×9.81) (1)
2
2
(129.329) (2×0.50)
=0.00176 (3 s.f.)
Experimental Value of Linear Mass Density of the Second Wire
µ(
𝑘𝑔
𝑚
0.00176+0.00180+0.00176+0.00174
4
)
= 0. 00176
Error Analysis
In the first part of the experiment, the measured frequencies for n=1, 2, 3, 4 were
131.065, 264.700, 403.475, and 537.109 Hz respectively. Based on the equation, 𝑓𝑛 =
𝑛
2𝐿
𝑇
µ
,
the mode number and normal mode frequency are linearly related/ or directly proportional to
each other (𝑓𝑛∝ 𝑛) when the length (L), tension (T), and mass per unit length (µ) of the wire are
kept constant. It is shown in the experimental data that the resonant frequencies are close to
having a linear relationship with the mode number. However, there appeared to be a discrepancy
between the theoretical and experimental values of the fundamental frequency, with a percent
error of 5.20%. The theoretical fundamental frequency, which was 126.470 Hz, appeared to be
lower than the average of the experimental values, which was 133.046 Hz.
% 𝑒𝑟𝑟𝑜𝑟 =
133.046−126.470
126.470
× 100 = 5. 20%
The discrepancy might be the result of having greater wire tension than 29.4N (
3 × 1. 0𝑘𝑔 𝑜𝑓 ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑚𝑎𝑠𝑠 × 9. 8
𝑚
2
𝑠
) from tightening the wire too much, and/or placing
the bridges within a distance slightly smaller than 0.50 m. Since the four experimental values of
fundamental frequency all appeared to be higher than the theoretical value, a systematic error
might come from the uncertainties in the linear density of the wire. All of these possible sources
of error could have contributed to a greater fundamental frequency than expected. In addition,
the experimental fundamental frequency calculated using the data of the lowest mode number
(n=1) appeared to have the smallest discrepancy from the theoretical value, and the values from
higher mode numbers became less and less accurate. Since signs from higher frequency modes
tended to dampen more quickly in the graph, inaccurate signals might have been picked out by
the experimenters, resulting in greater discrepancy in the higher order modes.
In the second experiment, two bridges on the sonometer were adjusted to vary wire
length while keeping the tension, mode number, and mass per unit length of the wire constant.
Under this condition, the fundamental frequency is expected to be inversely proportional to the
wire length (𝑓𝑛∝
1
𝐿
) . As expected, the graph based on the experimental data revealed that as the
wire length increases, the fundamental frequency decreases, shown in Figure 3. The graph may
not appear to be the perfect presentation of inverse proportionality because it was only based on
four data points.
Figure 3. Fundamental frequency vs. length of the wire with fixed tension, mode number,
and mass per unit length of the wire
During the third experiment, the experimenters varied the wire tension by placing the 1kg
mass into five different notches on the tensioning lever and measured the corresponding
fundamental frequency. With constant values of wire length, mode number, and mass per unit
length of the wire, it is expected that fundamental frequency is directly proportional to the square
root of tension (𝑓𝑛∝ 𝑇). Indeed, the wire tension increased as the hanging mass was moved
further down from the lever pivot, and with higher tension, the measured fundamental frequency
also increased. The graphical representation in Figure is based on the experimental data,
displaying a trendline similar to the function 𝑦 = 𝑥 .
Figure 4. Fundamental frequency vs. wire tension with a fixed length, mode number, and
mass per unit length of the wire
Conclusion
All desired results for the experiment were attained, and this collection of experiments
allowed the experimenters to investigate how fundamental frequency is related to length, mode
number, tension, and mass per unit length of a wire.
In the first experiment, the experimental value of fundamental frequency was higher than
the theoretical value. Possible errors might arise from the incorrect measurement of the distance
between the two bridges, the non-horizontal alignment of the tensioning lever, as well as
uncertainties in the linear density of the wire. In addition, it was possible that wrong signals for
higher frequency modes were picked by the experimenter since they were harder to spot in the
graph. All of these errors could affect the result of the experiment, leading to a higher
fundamental frequency than expected.
Moreover, the graphs based on the data obtained in the second and third experiments
depict the expected relationship between fundamental frequency and wire length and tension
respectively when other variables are kept constant. To obtain a more accurate graphical
representation, more data points could be taken. Overall, the experiments can be improved by
running the experiment multiple times to develop better judgment in picking out the correct
signals of string oscillations and having more trials for each experimental condition in order to
obtain a more precise average of the experimental data.
Questions
1. In the first 3 experimental sections, what was the motivation for the combination of
parameters in the last column? Explain.
○ The motivation for the combination of parameters in the last column in the first
three experiments was to get a sense of the precision of our experimental data.
This is because the value in the last column is expected to be a constant across all
the experimental conditions in each experiment. For example, in the first
experiment, the combination of parameters is shown as
𝑓𝑛
𝑛
, which is equal to
1
2𝐿
𝑇
µ
based on the formula 𝑓𝑛 =
𝑛
2𝐿
𝑇
µ
. Since wire
length (L), tension (T) , and mass per unit length of a wire (µ) were kept, the
values for
𝑓𝑛
𝑛
were all expected to have a constant value. In this way, if there is
any significant deviation from other data, the experimenters could easily spot it
and perform the experiment again if necessary.
2. To measure f1, why do we place the detector coil midway between the bridges?
What is the best location for the detector for higher order modes?
○ The detector coil was placed midway between the bridges because it was where
the antinode of a standing wave pattern is located. An antinode is a position where
a wave vibrates with a maximum amplitude, so placing the detector coil at that
position would allow the maximization of the signal, making it easier for
experimenters to pick out signals of wire oscillations from the noise in the
background. For higher order modes, the best location to maximize the signal
would depend on the wave pattern that one tries to capture. For the wave pattern
of an odd number harmonic, midway is still a good position to obtain the
maximum signal because there is an antinode at the midpoint for these harmonics.
However, this is not true for even harmonics. For example, for the wave pattern
for the second harmonic, the best location is at ¼ of the wire length (where the
two antidotes are). Taken together, one would need to examine the location of the
antinodes of a particular harmonic to determine the best location to place the
detector coil.
3. Explain why the string tension is given by 1-5 Mg as the mass is moved from notch 1
to notch 5.
○ According to the lab manual, the distance (d) between the lever pivot and the end
of the wire is the same as that between the pivot and notch 1. Suppose that the
lever arm is perfectly horizontal. When a hanging mass is hung onto notch 1, the
tension (T) created by the lever arm in the wire is then equal to the weight of the
hanging mass (Mg) based on this equation 𝑇 • 𝑑 = 𝑑 • 𝑀𝑔, which can be
rewritten as 𝑇 = 𝑀𝑔 after canceling out d on both sides.
○ By this logic, when the hanging mass is moved to notch 3 (shown below), the
string tension can be expressed as 3Mg because 𝑇 • 𝑑 = 3𝑑 • 𝑀𝑔, which can be
rewritten as 𝑇 = 3𝑀𝑔 after canceling out d on both sides. This is why string
tension is given by 1-5Mg as the mass is moved from notch 1 to 5.
4. If the tensioning lever was not horizontal how would it affect the experiment?
○ If the lever arm was not horizontal, then the effective moment arm would change.
As a result, the tension created by the lever arm would not be given by 1-5Mg as
the hanging mass (M) is moved from notch 1 to 5.
5. Why has f1 through f4 changed for the wire in section 5.4 changed compared to
0.022 inch wire?
○ This is because the wire used in 5.4 has a different mass per unit length than that
of the 0.022-inch wire. With a fixed length, tension mode number, resonant
frequencies are related to mass per unit length of a wire by 𝑓∝
1
µ
. Based on the
equation, the resonant frequencies (f1-f4) for the second wire are supposed to be
greater than those of the 0.022inch wire because the average value of mass per
unit length of the second wire (0.00176
𝑘𝑔
𝑚
) was found to be smaller than that of
the 0.022-inch wire.
○ To demonstrate how wires with different linear mass densities result in different
values of resonant frequencies, we suppose the resonant frequencies for the
0.022-inch wire are denoted by 𝑓𝑛, and the resonant frequencies for the second
wire is denoted by 𝑓'𝑛. Then, the ratio of the two frequencies can be expressed as
𝑓'𝑛
𝑓𝑛
µ
µ'
=
, where µ represents the mass per unit length of the 0.022-inch wire
and µ’ refers to the mass per unit length of the second wire. One can rearrange the
µ
µ'
equation into 𝑓'𝑛 =
𝑓𝑛, which shows that the resonant frequencies of the
second wire are directly proportional to that of the 0.022-inch wire by a factor of
µ
µ'
.
6. Work out equation 1 by using the equations for normal mode wavelengths and two
equations for the speed of wave.
λ
○ 𝐿 = 𝑛( 2 ), rearrange to get λ =
(normal mode wavelengths)
𝑇
µ
○ 𝑣=
○ 𝑣 = 𝑓λ, rearrange to get 𝑓 =
𝑣
λ
○ Then, replace v and λ with λ =
○ 𝑓=
2𝐿
𝑛
𝑣
λ
=
𝑇
µ
2𝐿
𝑛
𝑛
2𝐿
𝑇
µ
2𝐿
𝑛
and 𝑣 =
𝑇
µ