From the information given, the vibrating tuning fork is held near the mouth of a narrow plastic pipe partially submerged in water. This is a case of a closed air column. Recall, the resonance lengths of a closed air column are λ/4, 3λ/4, 5λ/4.
a) The wavelength of the sound produced by the tuning fork is found by the formula,
L1 = λ/4
where
λ is the wavelength
L1 is the first resonance length
By crossmultiplying,
λ = 4L1
From the information given,
L1 = 9 cm
we would convert it to meters
Recall, 100cm = 1
9cm = 9/100 = 0.09m
Thus,
λ = 0.09 x 4 = 0.36 m
The wavelength of the sound produced by the tuning fork is 0.36 m
b)To find the length of the air column for the second resonance, L2, we would apply the formula,
L2 = 3λ/4
substituting λ = 0.36 , we have
L2 = 3 * 0.36/4
L2 = 0.27m
the length of the air column for the second resonance is 0.27m
To find the length of the air column for the third resonance, L3, we would apply the formula,
L3 = 5λ/4
substituting λ = 0.36 , we have
L3 = 5 * 0.36/4
L3 = 0.45 m
the length of the air column for the third resonance is 0.45m
c) Since we know the temperature of the room, we would find the speed of sound, v by applying the formula
v = 331 + 0.59Tc
Given that temperature, Tc = 20, we have
v = 331 + 0.59 x 20 = 331 + 11.8
v = 342.8 m/s
Recall,
v = fλ
where
f = frequency
By making f the subject of the formula, it becomes
f = v/λ
Substituting λ = 0.36 and v = 342.8 into the formula, we have
f = 342.8/0.36
f = 952.22 Hz
the frequency of the tuning fork is 952.22 Hz