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| Resonance |
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| A mechanical system which is free to vibrate like a hacksaw blade clamped at one end, a diving spring board or the air in pipes has a natural frequency of vibration f0, which depends on its dimensions. When a periodic force of a frequency different from f0 is applied to the system, it vibrates with a small amplitude and undergoes forced vibrations. When the periodic force has a frequency equal to the natural frequency f0 of the system, the amplitude of the vibration becomes a maximum and the system is then set into resonance.
When the diver on the edge of a springboard begins to jump up and down
repeatedly, the board is forced to vibrate at the frequency equal to the
frequency of the jump. Initially, the amplitude is small and the board is said to be undergoing forced vibrations. As the diver jumps up and down to gain increasing height for his dive, the frequency of the periodic downward force reaches a stage where it is almost the same as the natural frequency of the board. The amplitude of the board then becomes very large and the periodic force is said to have set the board in resonance. |
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| If the prongs of a vibrating tuning fork are held on the top of a pipe, the air inside it is set into vibration by the periodic force exerted on it, by the prongs. In general, however, the vibrations are feeble, as they are forced vibrations. So, the intensity of the sound heard is correspondingly small. But when a tuning fork of the same frequency of the pipe is held over the pipe, the air inside is set into resonance by the periodic force and the amplitude of the vibrations is large. So, a loud note, which has the same frequency as the fork is heard, coming from the pipe and a stationary wave is set up. By achieving resonance in a pipe, the frequency of the fork or the velocity of sound can be determined. |
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| A narrow and long hollow tube of glass or a metal, open at both ends is taken. It is partially immersed vertically in a tall jar containing water. The length of the tube above the water level in the jar serves as an air column whose natural frequency depends on its length and the diameter of the tube. Using a suitable stand, the tube is clamped in a vertical position, so as to enclose a short length of air column. A tuning fork of a known frequency f, is excited by striking its prongs on a rubber pad and held horizontally above the upper end of the air column. The air column undergoes a forced vibration, resulting in a faint sound. The clamp is slightly loosened and the tube is gradually raised until, for a particular height, a loud sound is heard. Now the air column is in resonance with the tuning fork. The frequency of the tuning fork will be equal to that of the air column. At this stage, the clamp is tightened and using a metre scale, the height of the air column from the surface of water in the jar is measured. Let it be l1. This is called the first resonating length. The air column now, acts like a closed pipe and has the fundamental mode of vibration with a node at the surface of water and an antinode near the open end. |
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| There is a difference between the reflection of the wave at the closed end and at the open end. At the closed end, the reflection occurs exactly at that end and therefore, the node will be situated on the surface of water. At the open end of the tube, the air molecules will not have their maximum freedom of vibration because they are limited by the sides of the pipe. Thus complete reflection can occur, not exactly at the open end of the pipe, but a little outside it. Thus, the seat of the antinode will be a little beyond the open end. This additional distance between the end of the pipe and the point where complete reflection occurs, is called the end correction (e). Rayleigh determined this end correction to be 0.3d, where d is the diameter of the pipe. |
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| Using a slide calliper, the diameter of the pipe and hence, the end correction can be calculated. However, it can also be eliminated by finding the second resonating length as follows: |
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| The tuning fork is excited and held over the open end the tube again. The tube is raised further, till the sound intensity becomes maximum. This maximum sound, however, will not be as loud as in the previous case. The second resonating length l2 is measured. Now the air column will have the second mode of stationary wave formation with two nodes and two antinodes as shown in the figure. |
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| Then, |
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| Subtracting equation (i) from equation (ii), we get |
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| The velocity as determined above, gives the velocity of sound at the laboratory temperature toC. This temperature is noted using a wall thermometer in the laboratory. |
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| The experiment is repeated, using tuning forks of different frequencies. The readings are tabulated. The mean value of the velocity of sound at toC is found. If V0 is the velocity of sound at 0oC, then |
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| Thus, knowing the values of Vt and t, the velocity of sound at 0oc can be calculated. The readings are entered as follows. |
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| Temperature at which the velocity of sound is measured = t = ………. oC. |
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| Mean velocity of sound at toC = Vt = ………………… ms-1 |
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| Velocity of sound at 0oC = V0 = ………………… ms-1 |
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