The second resonant length of an open air column is 48cm. Determine the first and third resonant lengths
Well, isn't that a breath of fresh air? Determining resonant lengths for open air columns, huh? I hope you're ready for a wild ride of laughter!
Now, to determine the first and third resonant lengths, let's keep things simple. The first resonant length is half of the second resonant length. So, grab your math hat and divide 48 cm by 2. Ta-da! The first resonant length is 24 cm.
Now, for the third resonant length. Hold on to your seat, because this one's even crazier! The third resonant length is three times the first resonant length. Are you still with me? Good! Now, let's multiply 24 cm by 3 and find out that the third resonant length is a whopping 72 cm!
So, to recap, the first resonant length is 24 cm and the third resonant length is 72 cm. I hope these answers resonate with your sense of humor!
To determine the first and third resonant lengths, we can use the formula for the resonant lengths of an open air column:
L = (2n - 1) * λ / 4
Where:
L: Length of the resonant column
n: Resonant mode (1 for first, 2 for second, 3 for third, and so on)
λ: Wavelength of the sound wave
Given that the second resonant length is 48 cm (0.48 m), we can plug in the values and solve for the wavelength:
0.48 = (2 * 2 - 1) * λ / 4
Simplifying the equation:
0.48 = λ / 4
λ = 0.48 * 4
λ = 1.92 m
Now, we can calculate the first and third resonant lengths using the same formula:
For the first resonant length (n = 1):
L1 = (2 * 1 - 1) * 1.92 / 4
Simplifying:
L1 = 1 * 1.92 / 4
L1 = 0.96 m
For the third resonant length (n = 3):
L3 = (2 * 3 - 1) * 1.92 / 4
Simplifying:
L3 = 5 * 1.92 / 4
L3 = 2.4 m
Therefore, the first resonant length is 0.96 m (96 cm) and the third resonant length is 2.4 m (240 cm).
To determine the first and third resonant lengths, we need to know the formula for the resonant frequencies of an open air column.
The formula for the resonant frequencies (also known as the harmonics) of an open air column is:
f = (2n-1) v / 4L
Where:
f is the frequency of the harmonic
n is the harmonic number
v is the speed of sound in air
L is the length of the open air column
In this case, we are given the second resonant length (48 cm), so we can use this to solve for the frequency and then find the first and third resonant lengths.
Step 1: Calculate the frequency (f) using the given second resonant length (48 cm).
Given:
n = 2 (second resonant length)
L = 48 cm
Since we don't know the exact speed of sound in air, we can use the approximate value of 343 m/s.
v = 34300 cm/s
Substituting the values into the formula:
f = (2n-1) v / 4L
f = (2 * 2 - 1) * 34300 cm/s / (4 * 48 cm)
f = 3 * 34300 cm/s / 192 cm
f = 51450 cm/s / 192 cm
f ≈ 267.97 Hz
Step 2: Calculate the first resonant length using the calculated frequency (267.97 Hz).
Given:
n = 1 (first harmonic)
f = 267.97 Hz
Using the formula, we rearrange it to solve for L:
L = (2n-1) v / 4f
L = (2 * 1 - 1) * 34300 cm/s / (4 * 267.97 Hz)
L = 34300 cm/s / (4 * 267.97 Hz)
L ≈ 32.05 cm
Therefore, the first resonant length is approximately 32.05 cm.
Step 3: Calculate the third resonant length using the calculated frequency (267.97 Hz).
Given:
n = 3 (third harmonic)
f = 267.97 Hz
Using the formula, we rearrange it to solve for L:
L = (2n-1) v / 4f
L = (2 * 3 - 1) * 34300 cm/s / (4 * 267.97 Hz)
L = 5 * 34300 cm/s / (4 * 267.97 Hz)
L ≈ 80.11 cm
Therefore, the third resonant length is approximately 80.11 cm.
In conclusion, the first resonant length is approximately 32.05 cm, and the third resonant length is approximately 80.11 cm.
3•λ/4 =48
λ=64 cm
1• λ /4 =16 cm
5• λ /4=80 cm
Sorry. For open air column
λ =L2 =48 cm
L1 =λ/2 =48/2 =24 cm,
L3 = 3• λ/2=72 cm.