1. A stone is dropped from rest into a well. The sound of the splash is heard exactly 1.50 s later. Find the depth of the well if the air temperature is 10.0°C.
2. Calculate the sound level in decibels of a sound wave that has an intensity of 2.25 µW/m2
To solve these problems, we will need to use some equations and concepts related to the speed of sound and sound intensity.
1. Depth of the well:
The time it takes for the sound of the splash to reach us can be used to determine the depth of the well. Here's how we can do it:
The speed of sound in air at 10.0°C is approximately 331.4 m/s. However, sound travels slower in air as compared to a vacuum, due to factors such as air density and temperature. For simplicity, we can use an average value of 340 m/s as the speed of sound in air.
Let's assume the stone falls for time t and reaches the water at the bottom of the well. The sound then travels back up to reach us in 1.50 s.
The total time taken is the sum of the time it takes for the stone to fall (t) and the time it takes for the sound to reach us (1.50 s):
Total time = t + 1.50 s
Considering the speed of sound, we can write the equation as:
total distance = (speed of sound) × (total time)
Since the sound has to travel twice the height of the well (down and up), we can express the total distance as twice the depth of the well. Therefore:
2 × (depth of the well) = (speed of sound) × (t + 1.50s)
Simplifying the equation, we have:
depth of the well = (speed of sound) × (t + 1.50s) / 2
Now, plug in the given values for the speed of sound and the time delay of 1.50s to find the depth of the well.
2. Sound level in decibels:
The sound level can be calculated using the equation:
sound level (in decibels) = 10 × log10(intensity/1 µW/m²)
Given that the intensity of the sound wave is 2.25 µW/m², we can substitute this value into the equation to find the sound level in decibels.
sound level (in decibels) = 10 × log10(2.25 µW/m² / 1 µW/m²)
Now, calculate the logarithm and multiply it by 10 to find the sound level in decibels.
Note: In both cases, if your calculator does not have a log function, you can use the natural logarithm (ln) instead and convert it to base 10 logarithm using the conversion factor (log10(x) = ln(x) / ln(10)).