lab report

Student’s Name
Professor’s Name
Ishmael Essay
Date of submission
Vibrating Strings
Objectives
The of objectives of this experiment are
To establish experimentally between the fundamental resonance and the mass per unit of the string, length of the string and the tension in the string
To present an essential and useful graphical method for testing the quantities of x and y and their relationship are shown by single power function of the form y=a xn where the constant a and n can be found from the graph
To establish the relationship between resonant frequencies and higher order modes numbers experimentally
From the experimental results determine one general relationship and equation relating the resonant frequency with the four parameters, tension, mass per unit length, length and mode number. Finally, compare this experimentally found relationship with the theoretical expression
Introduction
In our day to day life experiences, we encounter many strings. Hence vibrating springs has become common. Most of the musical instruments across the world use the principle of vibrating springs when producing musical rhythms. Those who play such instruments know that when the tension of the string changes it will change the resonant frequency of vibration. Also when the thickness is adjusted the mass of the string will affect the rate. Length of the string also affects the sound which is being produced. A good example is base violin is much larger than the normal ones. This is what we what to investigate in this lab experiment.

Most waves for example water waves, and other waves are similar in behavior when they are transmitted through numerous boundaries and mediums. All the waves always reflect some energy are transmit some to the new medium .when dealing with a vibrating string the clamps at the end provide limits that waves are reflected from on end to another. When waves resent down a string of a given length, keeping the frequency constant the frequency being reflected and initiated will undergo constructive interference (Morse et al 66). This can be demonstrated using a stretched rubber band. Standing waves will always occur given rates when all the conditions are met which were stated above. We will now explore how resonant frequencies are affected by the four factors which are stated above.
Procedure
Equipment
A PASCO digital function generator, a ring stand, a ring stand, PC/SW750, electrical lead cables, strings
Methods and techniques
CASE 1: the Relationship between the Fundamental Resonant Frequency and the Length of the String
By using the medium mass string, eight the mass of the spring then the mass per unit length (kg/m) was calculated.
The string to the motor was tied then stretched to at least one meter to the pulley and a 200g mass hanged on the end hanging over the pulley.
The length of extra string hanging off the pulley was minimized. Where the length L of the string was the distance from the string tied to the arm of the vibrating motor to the top of the pulley where the string was held tightly.
This initial string length between the motor and the top of the pulley was the most extended length. The theoretical fundamental resonant frequency for five or six different string lengths including your initial length and four or five incrementally shorter lengths (e.g., L/4, L/3, L/2, 2L/3, 3L/4, L) was calculated.
The string was plucked with my finger near the middle point excites a vibration of the string primarily in its fundamental resonant mode of oscillation (also called the first harmonic). Then the string was also plucked to see how the vibration will look like.
The vibrating string enabled pumping the energy into the vibrating system at the same rate to what was being lost.
.To start, “auto” was clicked to change over to manual control making the “on” “off” buttons work. The calculated resonance frequency as the starting frequency was typed in. The voltage amplitude 5 volts was set. The frequency was changed in increments via. Data Studio
By setting the frequency of the vibrating motor slowly while watching the string, I was able to find the right frequency resulting in the fundamental resonant mode (“jump rope mode) for the string. For best results the speaker drive was adjusted until the “middle” of the resonance was found – that is, the frequency where the vibration amplitude is maximized.
CASE 2: the Relationship between Fundamental Resonant Frequency and String Mass/Length
By maintaining the string length at the maximum length which was used in Case 1, and the same mass was left on the end of the string.
The fundamental oscillation frequency for each of the string thicknesses at your station was calculated, thus varying the string’s mass per unit length (μ) were put into use.
The heavy then the light string were used. Remember that we already had one data point from your study of Case 1, the maximum length for all three string thicknesses.
using natural logarithms whether the relationship between fundamental frequency, f, and the mass per unit length, μ, that is f(μ) was determined, was simple power function (for our case it was the graph of ln (f) vs. ln (μ) would be a straight line).
The equation for the frequency being a function of mass/unit length was calculated.
Uncertainty-Graphical Analysis Supplement for details was referred to.
Experimental findings with the theoretical predictions were compared.
CASE 3: the Relationship between Fundamental Resonant Frequency and String Tension
Using the lightest string, and a length value the same as that used in Case 2, the tension Ts, was varied in the string.
The resonant frequency of the string for values of total mass which were being hanged on the end of the string of the following masses 50, 100, 150, 200, and 250g was calculated. The weight of the hanging mass was equal to the tension in the string, Ts. Data points done, from when the lightest string was used in Case 2 had already been done.
(Using ln (f) vs. ln (Ts)) the relationship between the resonant frequency, f, and the string tension, Ts, was graphically determined if it was a simple power function.
Again the uncertainty graphical analysis supplement was referred, and experimental results with the theoretical predictions were compared.
CASE 4: The Relationship between Frequency and Harmonic Mode Number 
The lightest string was used and the 200 or 250g hanging mass, set the length of the string at 1.5 m or higher.
The fundamental frequency was looked at the second harmonic (mode number n = 2) had a “jump rope” mode on each half of the string, but they oscillated in opposite directions. Increasing the oscillator frequency until the resonance was found and recorded.
The third harmonic had three “jump rope” modes on the string, etc. At the very least the data was collected, data for n = 1, 2, 3, and 4. The values of frequencies for even higher n values were determined.
The relationship f(n) between the resonant frequency, f, and the mode number, n, was determined by graphical means. Experimental findings with the theoretical predictions were compared.
Data
Thinnest string: 0.6g, 191cm.
Medium string: 1.7g, 165cm.
Thickest string: 8.5g, 210cm.
Case 1
The mass we are using is 200g, and the string we are using is the medium one.
Length(cm) Frequency(Hz)
141.5 17
136.5 17.8
131.5 18.7
126.5 19.4
121.5 20
Case 2
The mass we are using is 200g, and the string we are using is the thickest string.
Length(cm) Frequency(Hz)
141.5 7.9
Case 3
Mass(g) Frequency(Hz)
50 13.9
100 19.28
150 24.8
200 27.4
250 30.7
Case 4
The mass we are using is 250g, and the length is 150cm
Frequency(Hz) n
30.4 1
60.9 2
92.8 3
121.8 4
ANALYSIS
Graph 1

The graph presentation is inverse.
Graph 2

The first graph shows a negative gradient while the second one has a positive slope.
Case 2
The association between the rates and the mass per unit length was found and is was almost same the theoretical value.
Case 3
The experimental value in case 3 deviated from the hypothetical value.
Case 4
The experimental value is almost equal to the theoretical value.
Conclusion
To conclude, when it comes to string vibrations the displacement at the end was put at zero. A wave which is transverse moving along a fixed end will always be reflected in the opposite direction to that which the standard wave is traveling.
When a string is fixed at both ends of the clamp, it exhibited a strong response when only at resonance frequencies.

Will be the speed of waves which are transverse of string of length L. at other frequencies the string will not vibrate with much amplitude.

Work cited
Morse, Philip McCord, Acoustical Society of America, and American Institute of Physics. Vibration and sound. Vol. 2. New York: McGraw-Hill, 1948.