Experimental methods for the study of standing waves in strings
EE XPERIMENTAL METHODS FOR THE STUDY OF STANDINGWAVES IN STRINGS
A P
REPRINT
K. L. Cristiano
Facultad de Ciencias, Escuela de FísicaUniversidad Industrial de Santander (UIS)Colombia, Santander, Bucaramanga [email protected]
D. A. Triana
Facultad de Ciencias, Escuela de FísicaUniversidad Industrial de Santander (UIS)Colombia, Santander, Bucaramanga [email protected]
R. Ortiz
Departamento De Matemáticas Y Ciencias NaturalesUniversidad Autónoma de Bucaramanga (UNAB)Colombia, Santander, Bucaramanga [email protected]
A. F. Estupiñán
Departamento de Ciencias Básicas y HumanasUniversidad de Investigación y Desarrollo (UDI)Colombia, Santander, Bucaramanga [email protected]
September 30, 2019 A BSTRACT
Several cases of the explanation of the phenomenon of standing waves in strings, there are fewexperimental measurement tools when demonstrating this phenomenon in a classroom, it is for thisreason that we have implemented different forms to show how we can experimentally demonstratethe wave behavior of string vibration, where variations in the length, frequency and tension of thestring have been made, with the purpose of showing this phenomenon more generally and clearly forstudents.In this work, we present the step-by-step the implementation of two experimental procedures oflaboratory concerned with the study of wave strings, seeking as a primary objective to show studentsthe experimental way of obtaining and measuring the number of bellies in a string in addition ofbeing able to calculate both fundamentally and experimentally the fundamental frequency of eachstring by varying both its length and its tension using the resonance frequency.Finally, we present the respective experimental errors, for each of the assemblies and methods carriedout in the laboratory to perform the measurements shown and registered in this work. K eywords Fundamental frequency · Resonance frequency · Bellies · Nodes · Waves on strings.
Considering the importance in the course of Physics III that has the phenomenon of waves in strings and tubes [1], wewant to show in a clearer and more practical way this phenomenon, for example in the references [2–5]. We propose inthis work, the possibility of verifying both experimentally and analytically unknown string variables as for example:the volumetric and linear mass density which are intrinsic data of the material and we have calculated them indirectlythrough an experimental procedure shown in this document.We, the authors of this article, mainly want in this research work to make two experimental arrangements in which,for the first case we show a method, in which the tension of the string has been modified and the length has been leftconstant of the same and for the second part of this first case we have varied the length of the string and keep constantthe tension applied to the string. a r X i v : . [ phy s i c s . e d - ph ] S e p PREPRINT - S
EPTEMBER
30, 2019In the second case, we have used a strobe light, with which we want to use the definition of resonance to be able tomeasure the frequency of vibration of the string directly using a rotary motor, to which the input voltage is varied,which It allows us to have a control of the vibration frequency of the same [6, 7].After performing these two experimental methods, we have obtained the results, in order to indirectly find the linearmass density of each of the strings used in these two experimental arrangements. In order to show students two differentand alternative methods to calculate the linear mass density of the nylon strings used to perform these experiments.This paper is organized as follows: Section 2, describes the Theoretical study corresponding to the calculates for thetheory about of the waves in strings. In Section 3, shown the two experimental method procedure for the data takes. InSection 4, we show the experimental results and we compared the results obtained with this two methods presented inthis article.
When we want to model a wave phenomenon in physics, we must start from the second order differential equation ofthe propagation of a wave [8], which is posed in Equation (1) ∂ψ ∂t = (cid:0) v (cid:1) ∂ψ ∂x , (1)If, now we propose the function that models a simple harmonic oscillatory motion (See Equation (2)), which is obtainedas a solution to equation (1). ψ ( x, t ) = A · sin( kx − ωt ) (2)Now, taking into account the wave propagation equation in the string, we can write the wave equation, based on theTension T applied to it and the linear mass density (See Equation (3)), ∂ψ ∂t = (cid:18) Tµ (cid:19) ∂ψ ∂x , (3)Comparing the equation of motion (1) with (3), we have the value of the propagation velocity of the wave which is: v = Tµ , (4)The phenomenon physics of our system to study, consists in the behaviour of a stationary wave in a string fixed in itstwo extremes, how we shown in the Figure 1.It is important to note that in Figure 1, the value of n corresponds to the number of bellies reached as a function of thewavelength λ of the string. The equations, which govern the vibrational behavior of a string attached to its two ends,are given by the equations (5) and (6). v = λ n · f = 2 n L · f, (5) v = (cid:115) Tµ . (6)Where T is the tension applied to the string, lambda is the wavelength, f is the frequency of vibration, v the speed ofpropagation of the wave, and µ is the density of the line’s mass of the string for different values of n bellies [9]. Byrelating equation (5) and equation (6), we can obtain Equation (7), which directly relates the applied tension of thestring according to the frequency variation of the wave present in the string. T = 4 L f µn , ω n = (cid:115) Tµ · n · πL (7)2 PREPRINT - S
EPTEMBER
30, 2019Figure 1: Behaviour of a stationary wave, where we show the different bellies and nodes presents in the string. Thisimage was extracted from [6].
To carry out the experimental study of this work, we began to elaborate the assembly shown in Figure 2, correspondingto method 1.Figure 2: Experimental arrangement for the first method, where the main materials to be used are shown.In Figure 2, we can see that in this case, the experimental assembly that was performed with the purpose of measuringthe frequency of vibration of the string, we use the Software shown in Figure 2 that we have implemented, we haveclassified the data of as follows. 3
PREPRINT - S
EPTEMBER
30, 2019For the first part of method 1, we have varied the length of the string and keep the tension of the string constant asshown in Table 1.
L [m] f [ Hz ] f [ Hz ] f [ Hz ] f average [ Hz ] Mass [kg] Tension [ N ] f [ Hz ] f [ Hz ] f [ Hz ] f average [ Hz ] L = 2 . m , and thetension applied to it is varied, varying the masses placed at one end of the string (See Figure 2).It is important to remember that in tables 1 and 2 , the experimental data for the same string have been taken, which hasthe following characteristics: radius R = 7 . × − m , whose linear mass density is equal to . × − kg/m (corresponding to nylon) and its volumetric mass density is . × kg/m . The experimental setup that we have done for this method, we show in Figure 3.In method 2, we have used a strobe light, with which we can know the frequency of turning it on and off, in addition inFigure 3 the location of a rotation motor with variable rotation frequency is observed at which a variable voltage sourcehas been connected to it as can be seen in Figure 3.We collect the data shown in Table 3, where we have used the physical resonance phenomenon in which we take thefrequency that appears in the strobe light instrument; just in the instant when it emits the light at the same frequencythat the motor that makes the string vibrate.
Number of bellies n f [ Hz ] f [ Hz ] f [ Hz ] f average [ Hz ] . N , for the first five bellies.4 PREPRINT - S
EPTEMBER
30, 2019Figure 3: Experimental arrangement for the second method, where the main materials to be used are shown.Next, we take the experimental data of the period of oscillation of the string, varying the tension applied to it, keepingits length constant. The data obtained for this part, were recorded in table 4.
Tension [N] T [s] T [s] T [s] T average [s] f average [Hz] . m , for the first bellies, when we variate the tension of string. Starting from the data in Table 1, we can experimentally find the linear mass density of the string, if we plot on theY-axis the square of the length of the string and on the X-axis we graph the inverse of the square of the frequency, takinginto account that for this case, we have kept constant the tension applied to the string.Starting from the data in Table 1, we can experimentally find the linear mass density of the string, if we plot on theY-axis the square of the length of the string and on the X-axis we graph the inverse of the square of the frequency (SeeFigure 4), taking into account that for this case, we have kept constant the tension applied to the string. We obtainedEquation (8), from Equation (7). f = T n L µ (8)Using Data Table 2 and the Equation (7), we can relate the tension of the string according to the frequency obtained,keeping constant the value of the tension applied on the string. We can compare the frequency variation as the tensionin the string changes, based on the data taken in Table 2, we can obtain a graphical relationship of these results shownin Figure (5).From the Equation (8), if we clear the frequency of oscillation of the string, depending on the number of nodes generatedin the string, we can reach Equation (9). 5 PREPRINT - S
EPTEMBER
30, 2019Figure 4: Graph of the relation of the frequency of oscillation of the string in function of the variation of the length,keeping constant the tension T = . N .Figure 5: Graph of the tension in the string as a function of the frequency generated in the vibration of the string, for aconstant length of L = 2 . m . n = 4 L µf T (9)In the implementation of the second experimental method, we have obtained the data of the frequency and the numberof bellies, if we graph the frequency according to the number of bellies, we can experimentally calculate the linear massdensity µ of it (See Figure 6).Using now the data in Table 4, we can graph the string tension as a function of the frequency for the second method, weobey the relationship shown in Figure 7. 6 PREPRINT - S
EPTEMBER
30, 2019Figure 6: Graph of the number of bellies in the string as a function of the frequency generated in the vibration of thenylon’s string, for a constant length of L = 0 . m .Figure 7: Graph of the voltage as a function of the frequency, for a constant length of L = 2 . m .After performing this study, the following results were obtained for the calculation of the linear mass density µ of thestrings using both methods, (See Table 5). linear mass density Method 1 Method 2 µ (Theoretical) [ kg/m ] 5.057 × − × − × − × − µ (Experimental) [ kg/m ] 5.035 × − × − × − × − Table 5: Experimental results of the linear mass density µ of the strings, obtained from the data taken in the Physicslaboratory.Finally, in table 6, we have recorded the experimental errors obtained in our research work. Where we can compare theresults obtained in this experiment, for two different experimental methods taking into account that these have beenperformed, in order to obtain the value of linear mass density from indirect methods, which have been shown in thisarticle. 7 PREPRINT - S
EPTEMBER
30, 2019
Errors Method 1 Method 2
Absolute Error of µ [ kg/m ] 2.2 × − × − µ µ of the strings. In this work, we have been able to implement the measurement of linear mass density through two experimentalmethods in a physics laboratory; of two strings of different macroscopic characteristics, in which the implementationand generation of new laboratory materials has been carried out, in order to present two innovative proposals to measurethe value of µ using the wave phenomenon of vibration of the strings.Finally, in this article we have obtained very reliable results from our measurements, where we were able to registerexperimental errors below 10%, which guarantees us the reliability of the data taken and the certification of the goodresults obtained in these two experiments. In addition, we were able to implement the measurement of a physicalmagnitude of the string ( µ ) using two independent experimental methods. Acknowledgments
The authors would like to thank the Universidad Autónoma de Bucaramanga (UNAB), for lend us the installations andmaterials for carry to this experiment with which the analytical model presented in this paper could be validated.