In: Physics
13.18 Two piano strings with the same mass and the same length are to be tuned to 440 Hz (A on the equal temperament scale). One string, with a tension of 300 N is already tuned to 440 Hz. The second string frequency is 435 Hz. a) Should the tension of string 2 be increased or decreased to tune the frequency to 440 Hz? b) How much must the tension of string 2 be changed to change the frequency from 435 Hz to 440 Hz?
14.2 The wavelength of a 450 Hz sound wave in air has a wavelength of 0.78 m. a) What is the speed of sound in air? b) What is the temperature?
Given
two pianon strings of the same mass and the same length are to be tuned to 440 Hz
the string 1 tuned to 300 N of the frequency is f1 = 440 Hz
string 2 with frequncy 435 Hz , to tune the string2 to 440 Hz we have to increased te tension
because the frequency of the string fixed at both ends is
f1 = v/2l
and the speed of the wave on a string is v = sqrt(T/mue)
here mue is same so
f1 = (1/2l)(sqrt(T/mue))
that is frequency is proportional to the tension
f2/f1 = sqrt(T2/T1)
a)
so to increase frequency from 435 Hz to 440 Hz , we should increase the tension
b)
f2^2/f1^2 = T2/T1
440^2/435^2 = T2/300
T2 = 306.94 N
14.2
wavelength of sound wave in air is lambda = 0.78 m
frequency is f = 450 Hz ,
a)
the speed of sound in air is V = lambda*f = 0.78*450m/s = 351 m/s
b)
we know the formula is V = 331 m/s +0.61 m/s /C *T
T is the temperature
351 = 331+0.61*T
T = 32.786 0C