136 Experiment-367 F VERIFICATION OF THE LAWS OF VIBRATION OF STRETCHED STRINGS Jeethendra Kumar P K, Ajeya PadmaJeeth and Santhosh K KamalJeeth Instrumentation & Service Unit, No-610, Tata Nagar, Bengaluru-560092. INDIA. Email: labexperiments@rediffmail.com Abstract Using a new type of metal sonometer, speaker vibrator and power oscillator, the three laws concerning vibration of stretched strings are verified. This sonometer is easier to handle compared the available sonometers and moreover the vibrations produced are of high amplitude which are normally not observed in case of other sonometers conventionally used in physics laboratories. Introduction Marin Mersenne, a French scientist first published correct theoretical equations and formulated laws of vibration of stretched strings in 1636. He studied the vibrations of musical tones by using brass-ball weights attached to brass wires, adjusting the tension with appropriate weight attached to the wires [1]. Mersenne observed that the frequency of oscillation is inversely proportional to vibrating length of the string, directly proportional to the square root of tension and inversely proportional to the square root of mass of the string. Based on these observations, he formulated an equation describing vibration of stretched string as = 1 where ϑ is frequency of vibration, T is the tension (force) on the string, m is the unit mass of the string, and L is the vibrating length of the string. The tension T is applied by tying a weight (W) at the free end of the string. T = Wg 2 where W is weight tied to the string, and g is acceleration due to gravity.
137 Based on Equation-1, Mersenne formulated three laws of stretched string vibration as a) Law of velocity, b) Law of tension, and c) Law of mass These three laws were formulated as a result of his earlier observations on vibration of stretched strings. When the tension and mass per unit length are constant, the frequency is proportional to 1/L. ϑ α, or ϑl is constant. This is known as the law of Velocity. The constant appearing in this equation is the velocity of the sound through the wire which remains constant under this condition. Therefore, a graph of 1/L versus is a straight line, with slope given by Slope velocity = V = ϑλ where V is the velocity, and λ is the wavelength. From Equation-1 ϑl= gives the velocity, V, as V= and 3 L = λ 4 Keeping m and ϑ, constant, Equation-1 can be rewritten as ϑ= or =...5 Equation-5 represents a straight line, with slope given by Slope mass = 6 This is the law of Mass Keeping and T constant, one obtains the law of tension for a stretched string as ϑ m = T 7
138 Equatuion-7 represents a straight line with slope given by Slope Tension = T 8 This is the law of Tension. Therefore, one can summarize the three laws concerning vibration of stretched string, as given in Table-1. Table-1 Law of Held constant Proved constant Straight line Slope Velocity m, T υ L = V 1 versus ϑ Mass m, ϑ υ/ T = Tension ϑ, T υ m = T Stretched string vibrators V versus υ/ T 1 versus m T The three laws of vibration of stretched strings A thread or a wire stretched fully with a weight hanging on its end is called a stretched string. Such a string can be made to vibrate with external excitation. One can make the string vibrate with different types of vibrators. A tuning fork is the simplest type of exciter. The sound produced by the tuning fork travels through the string with the velocity which is governed by (i) the weight of the string, (ii) tension in the string, and (iii) frequency of the exciter. An AC current when passed through a copper wire held as a stretched string and pair of magnets placed on both the sides produce vibration in the string. An electromagnet placed above the stretched string can also produce vibration. If the wavelength of the exciter frequency matches with the length of the wire, vibrations can be seen as wave motion on the string. Based on this principle, some of the vibrators generally used in Physics labs are: Melde s tuning fork, electrical vibrator, and a set of tuning forks. In this experiment, a speaker vibrator is introduced which is much superior to the available vibrators because it can produce oscillations with large amplitude. The new metal Sonometer Sonometer [2] is an instrument generally used to verify the three laws of Mersenne, in which the sound box is designed using a wooden board. Sonometers made of teak wood are excellent but are not available now due to their high cost. Due to poor quality of wood used for making sonometers, they usually get damaged during transit and/or storage. Further, their long size (1m) adds to the possibility of their damage. Hence we have replaced wood with metal box of small size; the maximum size being 60cm. Smaller length together with lower weights and thin wire make them more suitable for frequencies between 30-130 Hz. In general, in the sonometer experiments a tuning fork or an electromagnet is used as the exciter. In this experiment we have used a tweeter (speaker) as the exciter. m
139 Tweeter vibrator Figure-1: Tweeter vibrator A three inch (7.5 cm) tweeter is selected for this application with 16 Ω dc resistance. The cap of the voice coil is reshaped. A light weight, 15mm long aluminum tube with 10mm dia is glued to the voice coil cap. A rubber rod is fixed on the aluminum tube. An aluminum strip, 10mm wide 1mm thick and 10cm long is fixed such that it remains in contact with the rubber rod, as shown in Figure-1. This picks up the vibrations from the voice coil and starts vibrating. A cotton thread or metal wire is tied to the aluminum plate which passes over a pulley and two bridges that are used to adjust the vibrating length of the string. Figure-2: The new sonometer with vibrator The entire unit is fitted to one end of the sonometer as shown in Figure-2. A power oscillator with variable frequency is provided along with the instrument which can be connected to the speaker. The power oscillator generates sinusoidal waveform; with frequency varying from 50Hz to 130Hz. Figure-3 shows the power oscillator. Figure-3: Variable frequency power oscillator
140 B Amplitude B Input sine wave E C A Equilibrium position A 0 T/4 T/2 3T/4 T C E Time Voice coil movement D D Figure-4: Vibrations of the voice coil (left) and the input sine wave (right) A sine wave of period T drives the speaker; the voice coil of the tweeter moves up and down as shown in Figure-4. When T=0, the voice coil is at its equilibrium position A. When the time is T/4, the voice coil moves to the upper position B; when time is T/2, the voice coil comes to the equilibrium position; when time is 3T/4 the voice coil is at the position D; and when the time is T the voice coil comes back to its equilibrium position. Hence during the period T, the voice coil completes one oscillation and, therefore, the period of the voice coil is the same as that of the sine wave. B Metal strip A,C,E Loop on string Speaker D Figure-5: Vibration of the string A string or thread tied to the vibrator will oscillate, as shown in Figure-5. There will be a loop with the maximum amplitude when the wavelengths of the vibrating string match with the input frequency in which case the vibrating length of the string is given by L= λ 9 where L is length of the loop vibrating with the maximum amplitude, and λ is the wavelength of the input signal driving the speaker
141 Hence if we determine the length of loop with the maximum amplitude, the velocity of the sound travelling on the string can be determined and the three laws of vibration of stretched strings can be verified experimentally [2]. Apparatus used Sonometer fitted with tweeter vibrator, power oscillator 50-130Hz variable frequency, slotted weights 5 numbers each of 20gm, and wires of iron (0.3mm dia), cotton thread, twin thread, nylon thread, and digital scale to weigh up to 300gm. The complete experimental set-up is shown in Figure-2. Experimental procedure The experiment consists of three parts, namely Part-A: Law of Velocity Part-B: Law of Tension Part-C: Law of Mass Part-A: Law of Velocity In this part of the experiment, m and T are held constant throughout the experiment, frequency of vibration is varied and the corresponding vibration length is determined. 1. The twin thread (green color) is selected for this part of the experiment. Thread of exactly one meter length is taken and its mass (M) is determined using the digital balance. Length (l) = 1m Mass (M) = 1.62 gm = 1.62x10-3 kg Mass per unit length (m) = =. =1.62210 kg/m 2. One end of the thread is tied to the tweeter vibrator and the other end is passed over the pulley and tied to a 3x20gm (W) slotted weight hanger. Hence the tension in the string T = Wg = 0.060 x 9.8 = 0.588 N 3. The power oscillator is now connected to the tweeter vibrator and the frequency is set to 130Hz. υ = 130Hz 4. The two bridges are brought to the center, taken apart very slowly and vibration of the string is observed. The distance between the strings is adjusted such that the string vibrates with the maximum amplitude with the single loop as shown in Figure-6. The length of the vibrating loop is noted from the graduated scale provided on the sonometer.
142 Figure-6: String vibration with the maximum amplitude Length (L) = λ = 14.4cm = 0.144m V = λυ = 0.144x130 =18.72 m/s This value can be verified using Equation-3. T V = =. m. = 19.05m/s The values obtained are tabulated in Table-2. Weight W (kg) Velocity (m/s) Table-2 Frequency υ (Hz) Vibrating length L=λ (m) Velocity V=υλ (m/s ) 0.060 19.05 130 0.144 18.72 110 0.163 17.93 90 0.198 17.82 76 0.238 18.10 70 0.261 18.27 62 0.290 17.98 50 0.364 18.20 Average (V) 18.15 Verification of the Law of Velocity 5. The experiment is repeated by varying the frequency to 110Hz, 90Hz etc. up to 50Hz. For each length of the vibrating loop, the maximum amplitude is determined and presented in Table-2. 6. A graph is drawn taking 1/L along the X-axis and frequency (υ) along the Y-axis, as shown in Figure-7. From the straight line graph, the slope is calculated which gives the velocity of the sound wave in the string. Slope velocity =18.85m/s. This agrees with the value obtained using equation-3
143 140 120 Frequency υ (Hz) 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 1/L (m -1 ) Part-B: Law of Tension Figure-7: Frequency variation with 1/L In this part of the experiment, m and are kept constant throughout the experiment. The vibrating length is determined for different values of tension in the string. 7. The same thread is used in this part of the experiment. 8. Frequency is set to 90Hz. 9. The 20 gram slotted weight hanger is tied to the thread and the vibrating length is determined υ = 90Hz m=1.62x10-3 L= λ = 12.3cm = 0.123m 10. The experiment is repeated by increasing the weight to 40, 60, 80 and 100 gm and for each case the vibrating length is determined and tabulated in Table-3. Table-3 Weight (gm) Tension T (N) T υ/ T L(m) 1/L 20 0.196 0.443 203.1 0.123 8.13 40 0.392 0.626 143.7 0.170 5.88 60 0.588 0.767 117.3 0.201 4.97 80 0.784 0.885 101.7 0.230 4.35 100 0.980 0.989 91.0 0.255 3.92 Vibrating length versus tension in the string 11. A graph showing the variation of υ/ T versus 1/L is drawn as shown in Figure-8. The slope of the straight line is determined.
144 250 200 ν/ T 150 100 50 0 0 1 2 3 4 5 6 7 8 9 1/L Figure-8: Law of Tension Slope tension = 25 = m= 0.04 m = 1.6 x 10-3 kg/m This value is in good agreement (1.62x10-3 ) with the value chosen for the experiment. Part-C: Law of Mass In this part of the experiment, the thread is changed, keeping the tension and frequency constant. Five different threads, of 1 meter length each, (three of cotton, one of copper and one of nylon) are provided with the sonometer. The mass per unit length of each one of the wires is determined by measuring its weight. 12. A 40 gram weight is tied to the end of the string and the frequency υ is kept constant υ =110Hz 13. The vibrating length of the string is determined L=λ=14.0cm = 0.14m Table-4 m (kg/m) x10-3 Vibrating length L (m) 1/L 1.62 0.04 4.40 0.14 7.14 0.91 0.03 3.30 0.192 5.2 0.65 0.025 2.75 0.217 4.6 0.38 0.019 2.09 0.285 3.5 0.28 0.014 1.54 0.384 2.6 Variation of the vibrating length with tension in the string
145 ν m 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 1/L Figure-9: Variation of with 1/L 14. The experiment is repeated by changing the thread, keeping the same tension and frequency. The readings obtained are tabulated in Table-4. 15. A graph showing the variation of 1/L with m is drawn as shown in Figure-9 and its slope is calculated Results Slope mass = 0.625 = T= 0.390 Actual T = Wg = 0.04 x 9.8= 0.392 The value agrees with the chosen value. In this experiment, Mersenne s laws of vibration of stretched strings are verified. In each case the constants appearing in the equations are determined and found to agree well with the corresponding theoretical value. The results obtained are tabulated in Table-5 Table-5 Law Expt. (Slope) Thet. Inference Velocity V= 18.85 m/s Velocity remains constant =19.05 m/s Tension m =1.6X10-3 Kg/m =1.62x10-3 Mass per unit length remains constant Mass T = 0.390 Wg = 0.392 Tension remains constant Experimental results References [1] http://www.justonic.com/mersenne.html [2] Jeethendra Kumar P K, Stretched string vibrator, LE Vol-4, No-4, Dec.-2004, Page- 303.