G11-6-2 PDF - Physics Past Paper

How does the velocity of a stationary wave formed in a string, with both ends firmly fixed, depend on the tension and mass per unit length of the string? The close Key Words: harmonic, tension, mass per unit length, overtone antin 6.3 RESONANCE COLUMN AND ORGAN PIPES Resonance and Resonant Frequency...

How does the velocity of a stationary wave formed in a string, with both ends firmly fixed, depend on the tension and mass per unit length of the string? The close Key Words: harmonic, tension, mass per unit length, overtone antin 6.3 RESONANCE COLUMN AND ORGAN PIPES Resonance and Resonant Frequency Several pendulums of different lengths are suspended from a flexible beam as shown in Figure 6.5. If one of them such as A, is set up into oscillation, the others will begin to oscillate because they are coupled by vibrations in flexible beam. Pendulum C, whose length is the same as that of A, will oscillate with the greatest amplitude since its natural frequency matches that of pendulum A which provides the driving force. TI F C D t Figure. 6.5 Example of resonance The amplitude of the motion reaches a maximum when the frequency of driving force equals the natural frequency of the system fo. Under this condition, the system is said to be in resonance. fo is called the resonant frequency of the system. Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving force. Th Us fre SO 72 Resonance Column If a vibrating tuning fork is placed over the open end of a glass tube partly filled with water as shown in the Figure 6.6, the sound of the tuning fork can be greatly amplified under certain conditions. The length of air column in the tube can be varied by raising (or) lowering the water level. At a certain length of air column, the loud resonant sound will be heard from the tube. The resonant sound will be heard at certain different length of air column. The wave is sent down the air column in the tube and it is reflected upwards when it hits the water surface. Once again it is reflected downwards when it reached the source. If the air column is just the proper length the reflected wave will be reinforced by the vibrating source as it travels down the tube a second time. In this way the wave is reinforced for a number of times and resonance is obtained from these multiple reinforcements. tuning fork tuning fork Figure 6.6 Adjustable resonance tube The tube shown in Figure 6.7 will have an antinode near the open end and a node at the water surface. The water at that end will not allow the air molecules to move downward. So they cannot move at the closed end and a node will be formed. At the open end the air molecules can move out freely and an antinode will be formed at the open end. Resonance can only be produced under the situation where a node is formed at water surface (closed end) and an antinode is formed at the open end. The velocity of sound can be found using resonance phenomena. Figure 6.7 shows the resonance phenomena at the length of air column 1, and 2. Since the antinodes lie just beyond the end of tube, correction c is added to the length of the air column. 14 Therefore, 4+c=2 32 and +c= 4 Figure 6.7 First and second resonance Using these equations, the wavelength of vibrating air column is λ = 2(1-4). At resonance, the frequency f of the tuning fork is equal to the frequency of vibrating air column. Thus, the velocity of sound is v=f2. 73 Organ Pipe The resonance phenomenon also occurs in an organ pipe. The organ pipes produce sound from the vibrations of air column in a pipe. Organ pipes are two types, closed organ pipe and open organ pipe. In closed organ pipe, one end is opened and another end is closed. For example whistle is a closed organ pipe. In closed organ pipes, an antinode exists near the open end (blowing end) while a node is formed at the closed end. The resonant frequencies for a closed organ pipe are as shown in Figure 6.8. The resonance single node in t stationary wave A A IN 1=4, 2=41 First Harmonic (fundamental) 4 322 41 1= 23. Third Harmonic (first overtone) 4 3 525 41 1=- 25 Fifth Harmonic (second overtone) 4 5 A A A A 72 1= 4/ Seventh Harmonic (third overtone) 4 7 Figure 6.8 Resonance phenomena in closed organ pipe The wavelength of the nth harmonic for vibrating air column in closed organ pipe is 41 (n = 1, 3, 5...) n We can now easily find the corresponding resonant frequency. Frequency of closed organ pipe is (6.6) nv fn= 41 (n = 1, 3, 5...) (6.7) where v = velocity of sound For the first harmonic, n = 1 ν 41 For the third harmonic, n = 3 3v 3f 41 For the fifth harmonic, n = 5 5v 5f 41 Closed organ pipe produces only odd harmonics. Therefore, third harmonic and fifth harmonic are called first overtone and second overtone respectively. 74 A The wavelengt The frequencie A flute can b at one end. Beats Beats are the p different frequ The number o sources. fb=f2~fi MM Textbook The resonance phenomena in open organ pipe are shown in Figure 6.9. The stationary wave with a single node in the open organ pipe corresponds to fundamental frequency (first harmonic). Thus, the stationary wave with two nodes is second harmonic (or) first overtone. A flute is an open organ pipe. A 1= 21 = 21 First harmonic (fundamental) 21 A 1= λ= Second harmonic (first overtone) 2 2 322 21 A 1= Third harmonic (second overtone) 2 3 42 21 A A A 1= 24 Fourth harmonic (third overtone) 2 4 Figure 6.9 Resonance phenomena in open organ pipe 21 The wavelength for the nth harmonic 2, = (n = 1, 2, 3...) n The frequencies of vibrating air column in open organ pipe are, ην f= 21 (n = 1, 2, 3...) (6.8) (6.9) A flute can be modeled as a pipe opens at both ends, while clarinet can be modeled as a pipe closed at one end.