(1) X = n (λ / 2) n = 0, 1, 2…n, where n can be any real number
(2) X = n (λ/ 4) n = 1, 3, 5…2n+1 where n can be any real number
In a string that is fixed on both ends, no vertical oscillation will take place at the points of nodes. However, for the rest of the points in the medium of the wave there will be the same frequency, and vertical displacements will differ for each point throughout the medium of the wave.
Figure 3:
The string can have several patterns of oscillation. Each pattern is unique in its own form, and there is a different frequency for each pattern. These different patterns of oscillation are referred to as normal modes or harmonics. The length L of the string between the fixed ends of each different pattern of oscillation is given by the formula:
L = n (λ) / 2 where n ≥ 1
The fundamental mode of vibration or the first harmonic occurs when n = 1. The second harmonic occurs when n = 2, the third harmonic when n = 3 and so on. Also it is important to note that ƒn = nƒ1 where ƒn is the frequency for each different mode specified by the number n ,and n is any real number which is multiplied by the fundamental frequency. For example ƒ2 = 2ƒ1, ƒ3 = 3ƒ1.
Figure 4: When n = 1
Figure 5: When n = 2
Figure 6: When n = 3
The speed of a standing wave is related to its wavelength and frequency given by the formula:
ƒ= frequency
Speed of the wave traveling along a stretched string is also dependent on the tension in the string, and the mass of the string per unit of length.
T = Tension, µ = mass of the string per unit of length
Putting both of these equations together and solving for the frequency you obtain:
f = (1 / λ)
When all of these formulas are combined and in particular when λ = 2L / n, L being the Length of the string and n being the number of modes, it is then possible to solve for each of the resonant frequencies.
ƒn is the frequency of each mode where n can be any real number.
ƒn = (n / 2L)
Procedure and Observations:
Required Apparatuse:
- Variable Frequency Vibrator
- Weight (100g, 200g, 300g)
- String
- Pulley
- Ruler
- Wooden Bridge
- Electrical Balance
Figure 5: The experimental setup
The Standing Waves apparatus was setup as shown above. A string with a length of 99.8cm was set up and attached to a vibrator from one end, and was around a pulley with a mass on the other end, and the string was attached to a rigid support, so the string could be set into resonance by the vibrator. The frequency of the vibrator was adjusted by turning the knob, so a standing wave pattern could be obtained. The frequency of the vibrator was increased to get a standing wave of higher modes of vibration. First a mass of 98.7 gr was experimented to see the highest mode of vibrations that could be obtained. This process was continued for a mass of 190.5 gr and 289.2 gr. A graph was then plotted where it was ƒ vs. n. Following the first method, the slope was calculated for the line of the best fit. Then the slope was put into the formula µ = (2LxSLOPE) -2 to get the µ and then the average of µ was calculated. In the second method µ was calculated using. At the end the calculated µ `s from two different approaches were compared.
Mass= 98.70 0.05 gr
Mass= 190.50 0.05 gr
Mass= 289.20 0.05 gr
Mass spring = 0.509 0.001 gr
Calculations, Graphs and Results:
1) Mass= 98.70 0.05 gr
Slope = = 22.41
= 1/ (2×22.41) ^2 = (4.97 0.05) ×10-4 Kg/m
2) Mass= 190.50 0.05 gr
Slope = = 22.19
=1/ (2×22.19) ^2 = (5.10 0.05) ×10-4 Kg/m
3) Mass= 289.20 0.05 gr
Slope = = 23.64
=1/ (2×23.64) ^2 = (4.47 0.05) ×10-4 Kg/m
The µavg could be calculated by the following formula:
(µ1+ µ2+ µ3) / 3 =>
{[(4.97 0.05) + (5.10 0.05) + (4.47 0.05)] ×10-4}/3 = (4.84 0.05) ×10-4 Kg/m
The second method:
In the second method the µ is calculated directly by the following formula:
µ = (mass of string)/ (length of string) which is ()
-
The mass of string is 0.509 0.001 gr = (5.09 0.001) × 10-4 Kg
-
Length of the string is 998 mm0.5mm = (9980.5)m
µ = (5.09 × 10-4)/ (998) = 5.10×10-4 Kg/m
Uncertainty in the measurement of in the second way:
=
∆ µ= 1.03 × 10-6 Kg/m
µ = 5.10×10-4 Kg/m 1.03 × 10-6 Kg/m
Discussion of Results And Conclusions:
It was observed that the mass of the string per unit of length () can be found by knowing the frequency of a string, tension and the mode of vibration. In this experiment the mass of the string per unit of length was measured with two different methods. In the first method, the mathematical formula was used to calculate the for different patterns of created standing waves. In this formula, slope can be found from the ƒ vs. n graph. Then the final result was found by calculating the average of obtained’s. The second method was measuring the µ directly by using the formula , where the mass of the string was determined by using the electrical balance. The average µ that was calculated in the first method was (4.84 0.05) ×10-4 Kg/m, and The value of by using the direct measurement was 5.10×10-4 Kg/m 1.03 × 10-6 Kg/m. These two results are close to reach other, but the result by using the second way is more accurate because of the fact that in the second method an electrical balance was used to measure the weight of the string which is relatively more accurate.
Since the mass of the string per unit of length values obtained through the both methods had minimum differences, this experiment was relatively a success. When finding resonance frequencies it was also known that the 2nd mode of vibration should be double the fundamental frequency. For example ƒ2 = 2ƒ1 and ƒ3 = 3ƒ1 and so on. Although the values obtained for higher modes of vibration were close to being multiples of one another, they were not exact. The causes of why these errors have occurred are due to a number of reasons. The air resistance could have affected the standing wave of the string which consequently affects the frequency of the wave. When the string reaches higher modes of vibration becomes more difficult to sense where the nodes are which will impact the frequency. There also could have been some errors in weighing and measuring the length of the string. To reduce the errors of this experiment it could have been performed several times.
References:
- PCS 125 Lab Manual (2008 fall), Department of Physics, Ryerson University, Toronto.
