PC1141 Physics I
Standing Waves in String
1
Purpose
• Determination the length of the wire L required to produce fundamental resonances
with given frequencies
• Demonstration that the frequencies f associated with fundamental resonances are proportional to 1/L where L is the length of the wire when resonances occur
• Determination of an experimental value for the mass per unit length of the wire µ
• Determination of an experimental value for the wave speed on the wire under a xed
applied tension T
• Determination of an experimental value for unknown frequency f from a linear least
squares t to the data L2 versus T
2
Equipment
• Sonometer
• A set of tuning forks
• Weights and hanger
• Transformer
• Rheostat
• Permanent bar magnet
3
Theory
Waves are one means by which energy can be transported. Waves in a string are an example
of a type of wave known as a transverse wave. The motion of the individual particles of the
medium (in this case the string) move perpendicular or transverse to the motion of the energy
that moves along the length of the string.
Consider the situation in Figure 1, which shows a string tied to a vibrator at one end that
then passes over a pulley at the other end with masses on the end of the string to provide
tension in the string. As the vibrator moves up and down at a xed frequency f , it causes a
wave of frequency f to propagate down the string.
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Standing Waves in String
Page 2 of 5
Figure 1: Experimental arrangement of vibrator and pulley.
The point at which the string passes over the pulley is a xed point and the wave is
reected from that point. Thus, the string is a medium in which two waves of the same speed,
frequency and wavelength are traveling in opposite directions. When these two waves interfere
with each other, a standing wave is produced, provided that the proper relationship exists
between the string length L and the wavelength λ of the wave.
When a standing wave is produced, its characteristic features are the existence of nodes
and antinodes at points along the string. A node N is a point for which there is no motion of
the string i.e., no displacement of the string from its equilibrium position. An antinode A
is a point on the string for which the amplitude of vibration is a maximum at all times.
Figure 2: First three standing waves for waves on a string.
The conditions for the formation of a standing wave are that a node N must occur at each
end of the string and that an antinode must occur between each pair of nodes. The distance
between two nodes is λ/2 or half of a wavelength of the wave. Therefore, in terms of the string
length L, a standing wave is possible when the following equation holds:
L=n
λ
2
(1)
where n = 1, 2, 3, 4, . . .. Figure 2 shows the rst three standing waves that are possible (i.e.,
n = 1, 2, 3). From the gure, it is clear that n can be thought of as the number of segments
of half wavelengths that are in each standing wave. Equation (1) can be solved to obtain the
wavelength for which standing waves can occur. They are
λ=
2L
n
(2)
where n = 1, 2, 3, 4, . . .. Each value of n determines what is known as a resonant mode of the
system. In particular, n = 1 is called the fundamental mode. In principle, there is no limit to
PC1141 Physics I
Semester I, 2007/08
Standing Waves in String
Page 3 of 5
the number of resonant modes of the system. In practice, it is very dicult to achieve better
than about n = 10 with the system used in the laboratory.
When the frequency of the vibrator is xed, the frequency of the waves on the string will be
xed. Thus, each dierent standing wave corresponding to each dierent n will have a dierent
wave speed v . The wave speed is determined by two other variables of the experimental
arrangement. They are the string tension T and the string mass per unit length µ. The
relationship between these quantities is given by
s
v=
T
µ
(3)
Using the relationship v = f λ, the expression for the speed v from equation (3) and the
expression for the wavelength from equation (2), leads to the following relationship for the
frequency f in terms of the tension T :
1
f=
λ
s
n
T
=
µ
2L
s
T
µ
(4)
Figure 3: Sonometer.
In this experiment, the sonometer consists of a long wooden sounding box as shown in
Figure 3. A steel wire is xed to a peg at one end of the box. It then passes over one xed
bridge A, one movable bridge B and a pulley, and is stretched by the application of a weight
W . The distance between two bridges determines the resonant length L of vibrating wire and
the tension in it is given by the applied weight W . The weights applied should not exceed
5 kg-weight to avoid accidents.
PC1141 Physics I
Semester I, 2007/08
Standing Waves in String
4
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Experimental Procedure
4.1
1.
2.
3.
4.
5.
6.
4.2
1.
2.
3.
4.
Experiment 1: Frequency and Resonant Length
Attach a hanger to the free end of the wire and apply weights of a total mass of 4.5 kg
to keep the wire taut.
Choose the tuning fork with the lowest frequency and record this frequency as f in Data
Table 1.
Sound the tuning fork and place its shank against the sonometer box to set the wire
vibrate.
Adjust the position of the bridge B from a position closer to the bridge A until the
resonance is taken place.
Hint: Place a light paper rider on the wire between bridges A and B to detect the
vibration of the wire. When a resonance is achieved, the rider will jump so violently
that it is thrown out from the wire.
Place the rider at the midpoint between bridges A and B , the rider would react most
strongly at this position if the distance between A and B corresponds to the fundamental
mode. Repeat a few times to obtain the best value for this resonant length. Record the
best resonant length as L in Data Table 1.
Repeat steps 25 with dierent tuning fork (dierent frequency) until EIGHT sets of
data are obtained. Record your data respectively in Data Table 1.
Experiment 2: Tension and Resonant Length
Attach a hanger to the free end of the wire and apply weights of a total mass of 100 g
to keep the wire taut. Record this value at m in Data Table 2.
Place a permanent bar magnet at the central portion of the wire such that the central
portion of the wire lies in the strong eld near one of the ends of the bar magnet.
Note: The direction of the magnetic eld and the current are at right angles to each
other. The magnetic force on the wire will be at right angles to both and will be reversed
in direction when the current is reversed.
Apply an AC voltage of 2 V from a main transformer to the wire of the sonometer
through a rheostat in series with the transformer.
Adjust the position of the bridge B from a position closer to the bridge A until the
resonance is taken place.
PC1141 Physics I
Semester I, 2007/08
Standing Waves in String
5.
6.
5
D1.
D2.
D3.
D4.
Page 5 of 5
Place the rider at the midpoint between bridges A and B to ascertain that this is the
fundamental mode of vibration. Measure the resonant length and record this value as
L in Data Table 2.
Repeat steps 25 with dierent amount of weights to keep the wire taut until FIVE sets
of data are obtained. Record your respective data in Data Table 2.
Note: The amount of weights to be used should be between 100 g and 700 g.
Data Analysis
Enter your data in the Data Table 1 into the Excel spreadsheet. Perform a linear least
squares t to the data, with the resonant length L as the y -axis and the reciprocal of
the resonant frequency 1/f as the x-axis. Determine the slope and intercept with the
corresponding uncertainties of the least squares t to the data.
Based on your results from the least squares t in D1, determine the experimental value
of the mass per unit length of the wire with the corresponding uncertainty. Hence, or
otherwise, determine the experimental value of the velocity of the wave traveling along
the wire with the corresponding uncertainty.
Enter your data in the Data Table 2 into the Excel spreadsheet. Perform a linear least
squares t to the data, with the square of the resonant L2 as the y -axis and the tension
of the wire T as the x-axis. Determine the slope and intercept with the corresponding
uncertainties of the least squares t to the data.
Based on your results from the least squares t in D3, determine the experimental value
of the frequency of the AC current with the corresponding uncertainty.
PC1141 Physics I
Semester I, 2007/08