Velocity of Longitudinal Pressure waves in Solid Bars
The purpose of the experiment is to compute the Young Modulus of specific rocks and metals by determination of the velocity of longitudinal pressure waves through the samples
Introduction
The speed of any mechanical waves of most material posses a unique relationship with both the elastic property of the material and the inertial property of the medium. The velocity of a mechanical wave is directly proportional to the square root of the quotient of elastic property and the inertial property.
V= Elastic propertyInertial property
There are three basic types of pressure waves; infrasonic waves, sound waves and ultrasonic waves. Infrasonic waves have frequencies below 20 Hz, sound waves have audible frequency range of between 20 Hz and 20,000Hz. Ultrasonic waves have higher than 20,000 Hz. Pressure waves travel as longitudinal waves in solids; however they can posses a transverse component (Serway 45).
This experiment exhibits the concept of pressure wave propagation through different samples. It employs both inertial property of the medium and the elastic properties to determine the velocity of the propagated wave. Longitudinal pressure waves were propagated through cylindrical samples of different lengths, L, of rock, glass and metal so as to evaluate the velocity of the pressure waves in the materials. A pulse generator is used to generate an electric square wave pulse of 10 microseconds (Serway 45). The square wave pulse is sent simultaneously to the first channel of the oscilloscope and to a transducer. The transducer is made of piezoelectric material and is linked to one end of the cylindrical solid bar. This connection is used to transform the electric square wave pulse to a pressure wave pulse which is then transmitted to the bar. The pressure wave travels all the way to the end of the bar.
The pressure wave through the bar is received using a second transducer connected at the other end of the bar. This transducer is used to convert the square pressure wave back to electric pulse wave. The converted electric pulse wave is then fed to the preamplifier for amplification. The amplified electric pulse wave is then relayed to the second channel of the oscilloscope. The oscilloscope is used to display two signals; it displays both the signal that is produced by the signal generator and the signal that passes through the bar. There is a time difference noted between the arrival times of the two singles (Serway 43). The difference is as a result of slower rate of pressure pulse propagation as compared to the electric pulse. The difference is significant in determining the velocity of the pressure wave in the bar. It employs the equation that describes the relationship between the length travelled by the pressure wave, L, the velocity and the time difference. The velocity of the pressure wave is directly proportional to the length of the bar and inversely proportional to the time difference. From this equation, the velocity of the pressure wave propagated through the bar can be determined.
v= L/Δt
The value of mass and volume of each of the cylindrical bars were used to obtain their density, P. On the other hand, the elastic property of the various samples used in the experiment is determined by Young Modulus, E. While elastic modulus is used to determine the fractional change in lengths when compression or tension is applied to a solid material, the concept is applied in customized to this experiment.
E= FL/ ΔLA
Where L is the length of the cylindrical bar, F is the cylindrical force, A is the cross sectional area of the bar and ΔL is the change in length of the bar. Density of the sample is considered as the inertial property (Serway 46). As a consequence, the velocity of the mechanical wave through the material is obtained as follows;
V= EP
Where v is the velocity of the mechanical pressure wave, E is the elastic property of the sample material and p is the density of the sample material.
Apparatus
The experiment used the following apparatus: rocks (greenstone, norite and granite), various materials (brass, steel, aluminum, glass and plastic), cradle, oscilloscope, piezoelectric transducers, vacuum grease, preamplifier, power supply, printer, balance, caliper, ruler and cables.
Procedure
The pulse generator, the oscilloscope and the preamplifier were checked to be sure to comprehend their connections, operations and settings. The wires and the cables connections of the setup were investigated for correct connections. The apparatus were used to carefully set up the experiment according to the schematics provided in the lab.
The dimensions and weight of the sample bars were measured and recorded appropriately. At least 6 bars available for the experiment were selected. The length and diameter of the selected bar were measured and recorded. The selected bars were weighed and their corresponding weights tabulated appropriately.
The oscilloscope, signal generator and transducers were used to measure the time delay between the direct and propagated signals. The oscilloscope was set to single trigger mode. A thin layer of vacuum grease was applied on the transducer’s membrane. The two transducers were brought head on contact and the pulse generator was used to send a pulse to the oscilloscope which was checked for signal display on the screen.
The thin layer of vacuum grease was constantly checked in case it needed re-application on the transducer’s membrane. The transducers were coupled to the ends of the sample bar to be measured. They were then held in tight contact with the ends of the bar and the pulse generator was switched on to send the electric square wave pulse. The arrival of the direct pulse and the propagated pulse were shown on channel 1 and 1 respectively. The time intervals and other setting of the oscilloscope were adjusted to increase the precision of determining the time difference between the arrivals of the two signals. The screen was printed for later use in precise determination of time difference.
Results
Calculations
The velocity of the propagated wave for different materials is as followed
Norite: V= 113188823434.54620563 = 5723. 711
Granite: V= 997707515.932.95053536 = 5502.68
Greenstone: V= 114172750329.06973136 = 6266.99
Steel: V= 264522911098.29962757 = 5187.47
Brass: V= 118370033294.05171734 = 3547.62
Glass: V= 89490258831.20626651 = 5355.11
Discussion
According to the results obtained on the velocity of the propagated pressure wave through different sample materials, it is apparent that the wave exhibit different velocity in each of the materials. This is a clear indication that different materials have different properties that affect pressure wave propagation through them. The velocity of the propagated pressure wave was recorded highest in greenstone (6266.99) followed by Norite, Granite, Glass, Steel and Brass respectively. These sample materials have different properties such as inertial properties and the elastic properties (Serway 48).The difference in these properties results in the difference in the velocity of the propagated wave as recorded. The velocity of the propagated pressure wave is affected by the inertial property (density) of the material and the elastic property of the material. Amplification of the propagated wave before it is fed to the oscilloscope is imperative so as to increase the precision of data gathering. The wave losses its amplitude as it propagates through different materials. The amplification helps in increasing the precision of readings of the signal displayed on the screen of the oscilloscope. There are different source of uncertainties exhibited in this experiment: meter stick with 0.5 mm uncertainty, caliper with 0.01 mm uncertainty and scale with 0.5 g uncertainty.
Works Cited
Serway, Raymond, and John Jewett. Physics for scientists and engineers with modern physics. Nelson Education, 2013.
