An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a 350 Hz tuning fork is held at the open end. Resonance is heard when the piston is at a distance d1 = 22.9 cm from the open end and again when it is at a distance d2 = 68.3 cm from the open end.
(a) What speed of sound is implied by this data?
(b) How far from the open end will the piston be when the next resonance is heard?
fn = nV/4L for container open at one end.
remember distance is in cm! therefore d1 = 0.229m.
384 = (1)V/(4)(0.229)
therefore:
V = 350 m/s
(a) To find the speed of sound, we can use the formula:
v = f * λ
where v is the speed of sound, f is the frequency of the tuning fork, and λ is the wavelength of the sound wave.
Given that the frequency of the tuning fork is 350 Hz, we need to find the wavelength. We know that the distance between the two resonances is equal to half a wavelength (since it corresponds to one complete cycle). Therefore, the wavelength can be calculated as:
λ = 2 * (d2 - d1)
Substituting in the given values:
λ = 2 * (68.3 cm - 22.9 cm)
λ = 2 * 45.4 cm
Now we can use the speed of sound formula to find the speed:
v = f * λ
v = 350 Hz * 2 * 45.4 cm
v ≈ 31820 cm/s
Therefore, the speed of sound implied by this data is approximately 31820 cm/s.
(b) To find the distance from the open end when the next resonance is heard, we can use the formula:
d3 = d2 + λ/2
where d3 is the distance from the open end when the next resonance occurs.
Substituting in the values we already know:
d3 = 68.3 cm + λ/2
d3 ≈ 68.3 cm + 45.4 cm/2
d3 ≈ 68.3 cm + 22.7 cm
d3 ≈ 91 cm
Therefore, the piston will be approximately 91 cm from the open end when the next resonance is heard.
To find the speed of sound implied by the given data, we can use the formula for the resonant frequencies of a closed-open cylindrical tube:
fn = (n * v) / (2 * L)
where:
fn is the frequency of the nth harmonic,
v is the speed of sound,
L is the length of the air column,
and n is the harmonic number.
In this case, we are given the frequencies (350 Hz) and the corresponding distances from the open end (d1 = 22.9 cm and d2 = 68.3 cm). We need to convert the distances into the length of the air column by subtracting them from the total length of the air column (which is the distance from the open end to the closed end).
(a) To find the speed of sound, we can set up the following equation:
(fn1 * L) / n1 = (fn2 * L) / n2
Substituting the given values, we get:
(350 Hz * (L - d1)) / 1 = (350 Hz * (L - d2)) / 2
Simplifying the equation gives:
2(L - d1) = (L - d2)
Expanding and rearranging the equation:
2L - 2d1 = L - d2
L = d1 - d2
Now we know the length of the air column, L. We can substitute this value and one of the given frequencies into the formula for the speed of sound:
v = (fn * 2 * L) / n
Picking one of the frequencies, let's say fn = 350 Hz:
v = (350 Hz * 2 * L) / 1
Now substitute the value of L:
v = (350 Hz * 2 * (d1 - d2)) / 1
Simplify the equation to find the speed of sound:
v = 700 * (d1 - d2) Hz
(b) To find the distance from the open end when the next resonance is heard, we can use the same formula:
(fn * L) / n = fn+1 * (L - x) / (n + 1)
Substituting the known values:
(350 Hz * L) / 1 = fn+1 * (L - x) / 2
Since we know L from part (a) and we are looking for x, we can rearrange the equation:
(350 Hz * L) / 1 = fn+1 * (L - x) / 2
Expanding and simplifying:
2 * 350 Hz * L = 1 * fn+1 * L - fn+1 * x
x = (fn+1 * L) / fn+1 - 2 * L
Now we can substitute the values we know to find x:
x = (fn+1 * (d1 - d2)) / (2 * (d1 - d2)) - (d1 - d2)
Simplifying the equation gives us the value of x.