when tuning a piano, amusician plays a note that should be at 110Hz while at the same time tapping a 110Hz tuning fork and holding it next to the strings. He hears beats at 4Hz.
(A) state and explain what frequencies the piano could be producing
(b) explain how the musician now finishes tuning his piano
(C) Draw a sketch graph to show the resultant sound as a variation in displacement against time. Label the time axis with values.
Yes
(A) To determine the possible frequencies the piano could be producing, we need to consider the phenomenon of beats. Beats occur when two sound waves with slightly different frequencies interfere with each other. The resulting sound is characterized by a periodic variation in loudness, known as beats. The frequency of the beats is equal to the difference between the frequencies of the two interfering waves.
In this case, the musician plays a note on the piano that should be at 110Hz and taps a 110Hz tuning fork, resulting in beats at 4Hz. This means that the frequency of the piano's note must be slightly different from 110Hz.
To find the possible frequencies, we can use the formula:
Beat Frequency (f_beat) = |Frequency of the piano note (f_piano) - Frequency of the tuning fork (f_tuning fork)|
Given:
f_tuning fork = 110Hz
f_beat = 4Hz
We can calculate the possible frequencies of the piano note by rearranging the formula:
f_piano = f_tuning fork ± f_beat
Substituting the values we have:
f_piano = 110Hz ± 4Hz
Therefore, the possible frequencies the piano could be producing are 106Hz and 114Hz.
(B) To finish tuning the piano, the musician needs to adjust the tension of the piano strings until the beats are eliminated. The objective is to make the frequency of the piano's note match precisely with the frequency of the tuning fork (110Hz in this case).
If the musician hears beats at 4Hz, this means that the frequency difference between the piano's note and the tuning fork is currently 4Hz.
To eliminate the beats, the musician needs to adjust the tension of the piano strings until the frequency difference is reduced to zero. This can be done by using the piano's tuning mechanism to tighten or loosen the strings until the desired frequency is achieved.
(C) I apologize, but as an AI text-based bot, I am unable to draw a sketch graph. However, I can explain how the graph would appear. The resultant sound, with beats at 4Hz, would create a periodic pattern of variation in displacement (amplitude) against time.
The graph would resemble a series of wave crests and troughs, where the distance between crests (or troughs) represents the displacement (amplitude) of the sound wave at a particular moment in time. The time axis would be labeled with values, such as seconds or milliseconds, indicating the progression of time.
The graph would show regularly spaced maxima (peaks) and minima (troughs) corresponding to the 4Hz beat frequency mentioned earlier. Each beat cycle would represent one complete variation from maximum to minimum and back to maximum displacement.
Please note that without actual values or specific frequency ranges, it is challenging to provide a precise graph. However, this general description should give you an idea of how the resultant sound as a variation in displacement against time would look with beats occurring at 4Hz.