Question: This is a monochord:
It is a instrument composed of a single resonating box and a single string. Different notes can be played by adjusting the tension in the string.
There are two monochords tuning against each other. Both have strings of 0.14 g/m linear mass density and 0.5 m length. One of them (Let's call it Instrument A) is tuned so that the tension in the string is exactly 100 N. The other's (Instrument B's) tension is unknown. Given that the beat frequency heard when both instruments are played is 15 Hz, what are the two possible tensions, in Newtons (N), of Instrument B?
Solution (assuming I didn't calculate anything wrong, which I might have):Known information: Both strings have a linear mass density of 0.00014 kg/m (don't forget to convert the units!) and a wavelength of 2*0.5 (if you don't know where I got the 2, go back to Khanacademy) = 1 m. Instrument A has tension of 100 N, and we're trying to find the tension of Instrument B.
There are two key equations here:
V = λf
V = sqrt(Ft / (m/L)) - Note that Ft is one variable: it's the force of tension. m/L is also one variable, in this case: its our linear mass density.
If we set the equations equal to each other, we get the most important single 21 characters of your life xD
λf = sqrt(Ft / (m/L))
If we plug in the numbers we know, we can find the frequency of Instrument A:
1 * f = sqrt(100/0.00014)
f = 845.15 for Instrument A.
Now that we know the frequency, we can simply add and subtract our beat frequency to get the two possible frequencies of Instrument B (once again, if you don't know why, go back to studying. Giancoli in this case is great)
The two possible frequencies are 845.15+15 = 860.15 Hz, and 845.15-15 = 830.15 Hz. Let's tackle them one at a time...
Working backwards using the same 21-character equation, 860.15 * 1 = sqrt(Ft / 0.00014)
Ft = 103.58 N
Working backwards the same way with the other possible frequency, 830.15 * 1 = sqrt(Ft / 0.00014)
Ft = 96.48 N
To double check our answer for reasonable-ness, just see that Instrument A's tension was 100, and the beat frequency is pretty small so our deviation from that shouldn't be too large as well. As you can tell, it isn't.
