Measuring the Velocity of Sound Applying a Standing Wave of a String - Lab Report Example

Name Course Institution Instructor Date Submission Date Abstract This report describes an experimental method that was used to determine the velocity of sound using a standing wave of a string. It involves use of various components including a vibrating device, a string, a set of weights, a tension measuring device, a tape measure and a top loading balance. Adjustments were made on the vibrating device to produce various standing waves with diverse wavelengths and various recordings such as weight used, frequency of the vibrator and the corresponding wavelength were taken. Similarly, with constant frequency of the vibrating device, various standing waves were determined and various recordings were taken. The results were graphically represented with the aim of ascertaining the velocity of sound in a string. Table of Contents Table of Contents 3 1.0 Introduction 4 1.1 Aims 4 2.0 Standing Waves 4 3.0 Materials and Procedure 6 3.1 Apparatus 6 3.2 Procedure 7 3.3 Results and Calculations 8 3.3.1 Part 1 8 3.3.2 Part 2 11 3.4 Discussions 12 4.0 Conclusions 13 5.0 References 14 1.0 Introduction The purpose of this experiment was to delve into standing waves on a string and to establish the relationship between the velocity of a wave in vibrating string and its characteristics including; mass per unit length, frequency and tension. A relationship exists between these characteristics and speed of a wave. Theoretically, the velocity of a standing wave on a string can be determined using the formula; Where; is the tension of the string is the velocity of standing wave is the linear mass density is the mass of the string is the length of the string This relationship can be determined experimentally and it was the main objective of the experiment. 1.1 Aims The purpose of this experiment was to: Establish the relationship between various characteristics of a vibrating string and velocity of the resultant standing wave. Develop research skills. An understanding of standing waves. Investigate the speed of sound in a string. 2.0 Standing Waves A wave is a travelling disturbance carrying energy as it propagates away from the source (Giordano, 2012: 384). Waves can be classified into either transverse or longitudinal. The vibration of particles for a transverse wave is perpendicular to the direction of propagation of wave while for longitudinal wave the direction of particle vibration is in parallel to the direction of propagation of the wave. Plucking a string under tension will result to production of transverse waves. If there is a continuous production of transverse waves from the source, they will be away at the other fixed end of the string. Interference occurs between the incident wave and the reflected wave when they meet and will result to a standing wave. Standing wave on a string Various properties of waves such as position of particles can be used to describe a transverse wave. The maximum distance above or below the equilibrium position of a wave particle travel in a transverse wave is referred to as amplitude (Giordano, 2012: 458). Wavelength () of a wave is the horizontal distance two successive points and it is the distance of one cycle of a wave (Giordano, 2012: 458). The time taken by one complete cycle of a transverse wave is referred to as period usually abbreviated as. The total number of cycles in one second is referred to as frequency usually abbreviated as. Frequency can also be obtained from reciprocal of period. These properties of waves can be used to determine the velocity of a wave. .............(i) In a single wavelength, the horizontal distance while change in time. Substituting these values in equation (i), then we shall have; or Therefore, the speed of a wave is dependent on wavelength, frequency and period. The velocity of a transverse wave on a string is also dependent on tension in the string and mass per unit length (properties of the string used) (Giordano, 2012: 390). From Newton’s second law, force is directly proportional to acceleration. Under high tension, a string will have its inter-particle force being very high. Therefore, from Newton’s second law tension which is a force will affect the velocity of a wave on a string. Velocity will be directly proportional to tensional force. Similarly, mass of the particles affects how it responds to the pulling force as a result of vibration. A string with greater mass per unit length will respond slowly thus moves with less velocity (Giordano, 2012: 460). Therefore, velocity of a wave is inversely proportional to mass per unit length of a string simply known as linear density. In point of fact, a direct proportionality exists between velocity and the square root of these quantities. This can be expressed as; Where; is the tension of the string is the velocity of standing wave is the linear mass density is the mass of the string is the length of the string 3.0 Materials and Procedure 3.1 Apparatus A vibrating device A string A set of weights A tension measuring device A tape measure A top loading balance 3.2 Procedure 1. Initially, familiarization with equipment used was done. 2. The oscillator was set to a frequency of 15 HZ. 3. The string was tensioned with a 200g weight. 4. The frequency of vibration was adjusted until a standing wave was produced as shown below. 5. Subsequently, the end of the vibrating blade was checked for presence of a node at the point of attachment to the string. A blade rattling against the case was an indication of a bad node. 6. The wavelength of the standing wave was then measured and recorded together with the corresponding values of weight used, tension in the string, frequency of the oscillator and wavelength of the standing wave. 7. Frequency of vibration was adjusted a different value and the same recordings were made. The procedure was repeated for five different standing waves. 8. The experiment was repeated with five different weights and recordings were made. 9. Finally, the string was weighed and its length measured. The values obtained were used to calculate the value of mass per unit length in kgm-1. 3.3 Results and Calculations 3.3.1 Part 1 200g Frequency (Hz) Tension (N) Wavelength, λ (m) (M-1) 15 2.2 1.6 0.625 20 2.2 1.2 0.8333 25 2.2 0.96 1.0417 30 2.2 0.8 1.25 35 2.2 0.6857 1.4584 100g Frequency (Hz) Tension (N) Wavelength, λ (m) (M-1) 15 1.3 1.2 0.8333 19 1.3 0.96 1.0417 23 1.3 0.8 1.25 27 1.3 0.6857 1.4584 31 1.3 0.6 1.6667 150g Frequency (Hz) Tension (N) Wavelength, λ (m) (M-1) 13.1 1.8 1.6 0.625 15.1 1.8 0.6857 1.4584 17.2 1.8 1.2 0.8333 22 1.8 0.96 1.0417 26.2 1.8 0.8 1.25 250g Frequency (Hz) Tension (N) Wavelength, λ (m) (M-1) 17 2.8 1.6 0.625 22.3 2.8 1.2 0.8333 28 2.8 0.96 1.0417 39 2.8 0.8 1.25 39.6 2.8 0.6857 1.4584 200g 100g 150 g 250 g 3.3.2 Part 2 Where; is the speed of sound in the string in ms-1 is the tension in the string in N is the mass per unit length (linear density) of the string in Kgm-1 Mass of the string is 13 g Length of the string 2.4 m Tension=2.2 = = 22.26 Tension = 1.3 = = 17.11 Tension = 1.8 = Tension = 2.8 = = 25.11 3.4 Discussions Comparing the values of part1 and part2, they vary by a very small difference. 200 g Part 1 part 2 percentage Error 23.56 22.26 100 g Part 1 Part 2 percentage Error 19.16 17.11 150 g Part 1 part 2 percentage Error 21.24 20.13 250 g Part 1 part 2 percentage Error 29.71 25.11 4.0 Conclusions From the above calculations, the percentage error is quite minimal. Therefore, there exists a relationship between the speed of a wave and properties of the string used. A part 1 calculation is the experimental method of determining this relationship. 5.0 References Giordano, N. College Physics, Vol 1 (ed 2). New York: Cengage Learning. 2012. Print. Read More