VibratingString

Vibratingstring

Theory

Fromthe literature, the phase of velocity c of any transverse wave of astring is given by the formula,where F is the tension in the string and isthe linear density of the string in kilograms per metre. The stringtension can be calculated from the mass M of the attached weight asF=Mg. for any wave, the phase velocity, wavelength alpha and thefrequency f are related by the equation.

c= fλ

Whenwe are substituting the equation and solving for the frequency gives.For an arbitrary tension and linear density, there will be noparticular relation between the wavelength and the length L of thestring among the vibrator and pulley. However, if we tune the systemby adding small masses to the hanger, we will eventually find a valueof F such that an integral number of half-wavelengths will fit intothe distance L. that is, nodes will occur spontaneously on both theleft and right ends and at several intermediate position as well.This condition is called resonance, and we say that standing waveshave been set up on the string. At resonance, the amplitude of astanding wave will be several millimeters. It is important to adjustthe hanging mass as closely as possible(+-g) so that the amplitude isa maximum. Suppose that the tension is such that n (standing) halfwaves have been set up in the distance L. as we increase the tensionfurther, we will eventually reach a new resonance condition where n-1half waves will fit in the length L. a further increase in tensionwill set up n-2 half waves. The resonance condition

L=n/2may be solved for 1/l and substituted into

Notethat as the mass of the hanger increase, the string stretches alittle, thus reducing the linear density u because the total mass isstretched out over a longer length. F we take r as the radius of thepulley, and l as the length of the string from the pulley to the masshanger, an approximate value for u will be

U=m/(L+nr/2+l)where m is the mass of the string.

Experiment

Thevibrator was set up and pulley on opposite ends of the table. Theradius of the pulley were measured and the distance L between thepulley and vibrator. A small piece of string about 20cm larger than Lwas cut. The mass of this piece of string was measured on the triplebeam balance that is sensitive to 0.01gram.

Thestring was then tied on one end to the vibrator, making a small loopat the other end, and after passing the string over the pulley put amass hanger through the loop. The system was tuned fro resonance byadding masses as needed until standing waves appear. After recordingl, m, and n increase the waves between the vibrator and pulley.

Discussion

Fromthe table below, error in frequency can be calculated fromstatistics.

Number of half waves |
wavelength |
Total mass M on hanger |
Tension in string |
Length string l |
Linear density |
frequency |

9 |
0.254 |
0.149 |
1.46 |
0.367 |
0.000246 |
303.8 |

8 |
0.285 |
0.177 |
1.74 |
0.368 |
0.000246 |
294.4 |

7 |
0.326 |
0.229 |
2.24 |
0.368 |
0.000246 |
293.1 |

6 |
0.381 |
0.315 |
3.09 |
0.369 |
0.000245 |
294.8 |

5 |
0.457 |
0.465 |
4.56 |
0.370 |
0.000245 |
298.6 |

4 |
0.571 |
0.732 |
7.17 |
0.373 |
0.000245 |
299.9 |

3 |
0.761 |
1.287 |
12.61 |
0.381 |
0.000244 |
299.0 |

Errorin frequency is normally given by the formula

Frequencyerror= measured frequency (fm)- signal frequency (fm)

=

=

Sourcesof error in the experiment were improper measuring of strings. Thiscan be a human error generated. One can measure or write wrongfigures that can result in wrong results. The second sources of errorare an improper measurement of frequency. It can be the values readfrom the machine are too high or low. Hence, producing a largerpercentage error that leads to wrong output.

Thirdsources of error in incorrect visual observation of the number ofloops, for instance, six long mistaken to five. When one does notobserve the correct measurement, he/ she will carry the error to thenext step. This will, therefore, result in the wrong verdict of theexperiment. The last sources of error that can emerge from theexperiment are the mistake in calculation

Amistake in the calculation can either be wrong multiplication,addition, and subtraction of the values. It can also be wrongdivision and forgetting that you have carried a certain number duringmultiplication. It, therefore, results in the wrong answer thataffects the conclusion and more so the aim of the experiment. It willrequire one to repeat the whole experiment which tends to be tediousand time wasting.

Heavyor thicker string will have an effect on the system. For instance, aheavy string of the same material lowers its frequency for a givenlength and tension. Just like a musical instrument. Therefore usingheavy string reduces frequency resulting in the stiffness thataffects vibration and amplitude of the system.

Inconclusion, it is important for one to consider all requiredapparatus are there. Secondly, should be cautious with errors thatmight arise during the experiment to avoid such errors fromoccurring.