19 cars in a circle at a boom box competition produce a 130 dB sound level at the center of the circle. What is the average sound level(dB) produced there by each, assuming interference effects can be neglected?
Please someone help me to solve this problem. Thanks a lot.
To solve this problem, we can use the inverse square law, which states that the sound intensity decreases with the square of the distance from the source.
Let's assume that each car produces the same sound intensity level (L) in all directions. Given that there are 19 cars in a circle, the sound from each car will spread out in all directions and add up at the center of the circle.
The total sound intensity (I_total) at the center of the circle is given by the sum of the individual sound intensities (I) from each car:
I_total = 19 * I
Using the formula for the ratio of sound intensities based on distance:
I_total / I_single = (D_single / D_total)²
Where D_single is the distance from one of the cars to the center of the circle, and D_total is the radius of the circle.
As the sound intensities add up linearly, we can equate the sound intensities at the center to find the average sound level (L_avg):
L_avg = 10 * log10(I_total / I_reference)
Where I_reference is the reference sound intensity (10^-12 W/m²).
Now, let's calculate the average sound level:
1. Determine the distance:
As the cars are arranged in a circle, we can assume that the distance from each car to the center is the same. Let's consider this distance as D_single.
2. Calculate the total distance:
The distance from the center of the circle to any car is equal to the radius of the circle. Let's denote this distance as D_total.
3. Calculate the total sound intensity:
Using the inverse square law:
I_total / I_single = (D_single / D_total)²
19 * I_single / I_single = (D_single / D_total)²
19 = (D_single / D_total)²
Taking the square root of both sides:
√19 = D_single / D_total
D_single = D_total * √19
4. Calculate the average sound level (L_avg):
Using the formula:
L_avg = 10 * log10(I_total / I_reference)
Since I_total = 19 * I_single:
L_avg = 10 * log10(19 * I_single / I_reference)
Now, you can substitute the values for I_single and solve for L_avg.
To solve this problem, we can use the inverse square law for sound intensity. According to this law, the sound intensity at a certain distance from the source is inversely proportional to the square of the distance.
The formula for the sound intensity is as follows:
I = P / (4πr²)
Where:
I is the sound intensity,
P is the power of the sound source, and
r is the distance from the source.
In this case, we have 19 cars producing a 130 dB sound level at the center of the circle. Since interference effects can be neglected, we can assume that each car produces an equal amount of sound.
To find the average sound level produced by each car at the center of the circle, we need to find the total power of the cars and then calculate the sound level based on that power.
The formula for sound level (dB) is:
L = 10 log(I/I₀)
Where:
L is the sound level in decibels,
I is the sound intensity, and
I₀ is the reference intensity, which is typically set at 10^(-12) W/m².
First, we need to calculate the total power of the cars. Since each car produces the same amount of sound, we can divide the total sound level (130 dB) by the number of cars (19) to get the sound level produced by each car.
L_car = 130 dB / 19 ≈ 6.842 dB
Next, we need to convert the sound level to sound intensity. Using the formula L = 10 log(I/I₀), we can rearrange it to solve for I:
I/I₀ = 10^(L/10)
Substituting the sound level produced by each car into the formula, we get:
I/I₀ = 10^(6.842/10)
I/I₀ ≈ 5.363
Finally, we can rearrange the sound intensity formula I = P / (4πr²) to solve for P:
P = I * 4πr²
Since we are interested in the sound intensity at the center of the circle, where the radius is 0, the formula becomes:
P = I * 4π(0)²
P = I * 4π(0)
P = 0
As we can see, the power of each car is zero since there is no distance between the car and the center of the circle. Therefore, there is no sound being produced by each car individually at the center.
To summarize, assuming interference effects can be neglected, there is no sound being produced by each car individually at the center of the circle.
L=10 log₁₀ (I/I₀)
I₀= 10⁻¹² W/m²
I/I₀ = 10^(L/10)
I= I₀•10^(L/10) =
=10⁻¹²•10^(130/10)=10 W/m²
I₁ = I/19=10/19= 0.53 W/m²
L=10 log₁₀ (I₁/I₀) =
=10 log₁₀ (0.53/ 10⁻¹²)=144.7 dB