An air column open at one end only is increased in length. The air column first resonates with a given turning fork when the length of the air column reaches 20 cm. The next two consecutive lengths that will resonate with the same tuning fork will measure.
To find the next two consecutive lengths that will resonate with the same tuning fork, we can use the formula:
L = λ/4
where L is the length of the air column and λ is the wavelength.
Given that the current length at resonance is 20 cm, we can calculate the wavelength at resonance using the formula:
λ = 2L
So the wavelength at resonance is:
λ = 2 * 20 cm = 40 cm
Now, to find the next two consecutive lengths at resonance, we need to consider the relationship between the length and the wavelength.
If we increase the length of the air column by one wavelength (40 cm), the next length at resonance will be:
L1 = 20 cm + 40 cm = 60 cm
Similarly, if we increase the length by another wavelength (40 cm), the next length at resonance will be:
L2 = 20 cm + 40 cm + 40 cm = 100 cm
Therefore, the next two consecutive lengths that will resonate with the same tuning fork will measure 60 cm and 100 cm.
To determine the next two consecutive lengths that will resonate with the same tuning fork, we need to understand the concept of the relationship between the length of an air column and its resonance frequency.
The resonant frequency of a column of air in an open-ended tube is determined by the length of the tube. The general formula for the resonant frequency in an open-ended tube is:
f = (n * v) / (2 * L)
Where:
- f is the frequency of the sound wave (resonance),
- n is the harmonic number (1, 2, 3, ...),
- v is the speed of sound in air (approximately 343 m/s at room temperature), and
- L is the length of the air column.
In our case, the first resonant length is given as 20 cm (or 0.20 m). Let's calculate the corresponding frequency of the tuning fork using the given length:
f₁ = (n * v) / (2 * L₁)
= (1 * 343) / (2 * 0.20)
= 1715 Hz
Now, we need to find the next two consecutive lengths. We can do this by setting up an equation using the relationship between the harmonic numbers and the resonant frequencies:
f₁ = (1 * v) / (2 * L₁)
f₂ = (2 * v) / (2 * L₂)
f₃ = (3 * v) / (2 * L₃)
Since f₁, f₂, and f₃ all represent the same tuning fork's resonance frequency, we can equate them:
f₁ = f₂ = f₃
Substituting the given values:
1715 = (1 * v) / (2 * 0.20)
1715 = (2 * v) / (2 * L₂)
1715 = (3 * v) / (2 * L₃)
Simplifying the equations, we get:
v / L₁ = 343 / (2 * 0.20) --- Equation 1
2v / L₂ = 343 / (2 * L₂) --- Equation 2
3v / L₃ = 343 / (2 * L₃) --- Equation 3
From Equation 1, we can solve for v / L₁:
v / L₁ = 343 / (2 * 0.20)
v / L₁ = 856.25
Now, we can substitute this value in Equation 2:
2 * (856.25) = 343 / (2 * L₂)
2 * L₂ = 343 / (2 * (856.25))
2 * L₂ = 0.20
Simplifying further, we get:
L₂ = 0.20 / 2
L₂ = 0.10 m (or 10 cm)
Similarly, we can find L₃ using Equation 3:
3 * (856.25) = 343 / (2 * L₃)
3 * L₃ = 343 / (2 * (856.25))
3 * L₃ = 0.30
Simplifying further, we get:
L₃ = 0.30 / 3
L₃ = 0.10 m (or 10 cm)
Therefore, the next two consecutive lengths that will resonate with the same tuning fork after a 20 cm length are 10 cm and 10 cm.