A 3580 Hz tuning fork is sounded at the same time that a note on a piano is struck. You hear 8 beats/s. You loosen the string slightly and now hear 12 beats/s. What is the frequency of the vibrating piano string now after the loosen ing?
you hear the difference frequency
8 = 3580 - f
f = 3472 original
12 = 3580 -f
f = 3568 final
good, the frequency went down when I loosened the string
so 3568 Hz
WHAT if I wrote the difference the other way (all I know is the absolute value)
8 = f - 3580
f = 3588
12 = f - 3580
f = 3592
oh my the frequency went up when we loosened the string - no way so the difference is the other way around
and
3568 is the answer
To determine the frequency of the vibrating piano string after loosening, we need to use the concept of beats. Beats occur when two sound waves with slightly different frequencies interfere with each other, resulting in an oscillating pattern of sound intensity.
Initially, the tuning fork has a frequency of 3580 Hz, and the piano string has an unknown frequency. We hear 8 beats per second, indicating that the frequency of the piano string is either 8 Hz higher or lower than the tuning fork.
After loosening the piano string, we now hear 12 beats per second. This change in beat frequency indicates that the frequency of the piano string has further shifted.
To find the change in frequency, we can calculate the difference between the two sets of beat frequencies:
Change in beat frequency = New beat frequency - Initial beat frequency
Change in beat frequency = 12 beats/s - 8 beats/s
Change in beat frequency = 4 beats/s
Since each beat corresponds to a difference of 1 Hz, the change in beat frequency of 4 beats/s means the difference in frequency between the tuning fork and the piano string has increased by 4 Hz.
To find the frequency of the piano string after loosening, we need to add or subtract this change from the initial frequency. Since we don't know if the piano string is higher or lower in frequency than the tuning fork, we consider both possibilities:
If the initial frequency of the piano string is higher than the tuning fork:
Frequency of piano string = Frequency of tuning fork + Change in frequency
Frequency of piano string = 3580 Hz + 4 Hz
Frequency of piano string = 3584 Hz
If the initial frequency of the piano string is lower than the tuning fork:
Frequency of piano string = Frequency of tuning fork - Change in frequency
Frequency of piano string = 3580 Hz - 4 Hz
Frequency of piano string = 3576 Hz
Therefore, after loosening the piano string, the frequency can be either 3584 Hz or 3576 Hz depending on the initial frequency relationship with the tuning fork.