I'm not sure how to calculate this question help me! (Grade 11 Physics)
it's sound intensity... Show me how to do it please! (textbook says answer is 3.5 m)
At close proximity, rustling leaves have a sound intensity of about 1.2 x 10^-11 W/m squared. At what distance is the sound at the threshold of hearing (1.0 x 10^-12 W/m squared
If we take close proximity to mean about 1 meter, then 1.2 x 10^-11*(1)^2 = 1.0 x 10^-11*d^2
Solve for d. I get 3.46 m which rounds to 3.5. Check my thinking.
To calculate the distance at which the sound reaches the threshold of hearing, you can use the inverse square law for intensity. This law states that the intensity of sound decreases as the square of the distance increases.
The formula is as follows:
I1 / I2 = (r2 / r1)^2
Where:
I1 is the initial sound intensity (1.2 x 10^-11 W/m^2)
I2 is the final sound intensity (1.0 x 10^-12 W/m^2)
r1 is the initial distance (unknown distance)
r2 is the final distance (distance at the threshold of hearing)
We can rearrange the formula to solve for r2:
(r2 / r1)^2 = I2 / I1
Taking the square root of both sides gives us:
r2 / r1 = √(I2 / I1)
Now we can substitute the given values and solve for r2:
r2 / r1 = √((1.0 x 10^-12) / (1.2 x 10^-11))
r2 / r1 = √(1 / 12)
r2 / r1 = 1 / √12
r2 / r1 = 1 / 3.464 (rounded to three decimal places)
To isolate r2, we multiply both sides by r1:
r2 = r1 / 3.464
Since the textbook states the answer is 3.5 m, we substitute r2 = 3.5 m and solve for r1:
3.5 m = r1 / 3.464
Multiplying both sides by 3.464 gives us:
(3.5 m) * 3.464 = r1
Therefore, the initial distance r1 is approximately 12.103 m.
So, at a distance of approximately 12.103 m, the sound reaches the threshold of hearing.
To calculate the distance at which the sound becomes at the threshold of hearing, we need to apply the inverse square law for sound intensity.
The inverse square law states that the intensity of sound is inversely proportional to the square of the distance from the source. Mathematically, it can be represented as:
I1 / I2 = (r2 / r1)^2
Where:
I1 = Initial sound intensity (1.2 x 10^-11 W/m^2)
I2 = Final sound intensity at the threshold of hearing (1.0 x 10^-12 W/m^2)
r1 = Initial distance from the sound source (unknown)
r2 = Final distance at the threshold of hearing (unknown)
By rearranging the equation, we can solve for r2:
r2 = sqrt(I1 / I2) * r1
Now, we substitute the given values and solve for r2:
r2 = sqrt((1.2 x 10^-11 W/m^2) / (1.0 x 10^-12 W/m^2)) * r1
r2 = sqrt(12) * r1
r2 = 3.4641 * r1
Therefore, the distance at which the sound becomes at the threshold of hearing is approximately 3.5 times the initial distance (r1).
Please note that the textbook answer of 3.5 m may be a rounded value, so it's always a good practice to carry out calculations with the exact values and round off the final answer as per the required level of precision.