Two fire alarms are placed in a building, one 2.2 m north of the other. A person sitting at a desk which is 52.9 m east and 9.4 m north of the point mid-way between the alarms does not hear them when they sound. What is the lowest frequency that the alarms can be sounding at assuming that the alarms are both perfectly in phase?
To solve this problem, we need to consider the concept of sound interference and the conditions for constructive and destructive interference.
First, let's visualize the given information. We have two fire alarms in a building, with one placed 2.2 m north of the other. The person sitting at the desk is located 52.9 m east and 9.4 m north of the point mid-way between the alarms.
Now, let's break down the problem:
1. Determine the path-length difference: Since the person does not hear the alarms, there must be destructive interference occurring at their location. In order to have destructive interference, the path-length difference between the two alarms should be an odd multiple of half a wavelength.
2. Calculate the path-length difference: The path-length difference can be calculated by finding the distance between the person and each alarm and subtracting them. Let's calculate it step by step:
- Distance to the first alarm (north): This distance is 2.2 m.
- Distance to the second alarm (south): To calculate this distance, we need to find the total distance between the person and the mid-point between the alarms. The distance north from the mid-point is given as 9.4 m, and the horizontal distance from the mid-point is given as 52.9 m. Considering these two distances, we can calculate the hypotenuse using the Pythagorean theorem: sqrt((9.4m)^2 + (52.9m)^2). This will give us the distance from the person to the second alarm.
3. Calculate the lowest frequency: We can use the formula v = f * λ, where v is the speed of sound, f is the frequency, and λ is the wavelength. Since the lowest frequency corresponds to the longest wavelength, we can rearrange the formula to find the wavelength: λ = v / f.
- Speed of sound: The speed of sound is approximately 343 m/s in air at room temperature.
- Wavelength: Substitute the values for the speed of sound and the lowest frequency into the formula to calculate the wavelength.
4. Convert the wavelength to meters: Since the distances in the problem are given in meters, we want to make sure the wavelength is also in meters. If it's given in another unit, we need to convert it to meters.
Once the wavelength is determined, we will have the answer to the question - the lowest frequency that the alarms can be sounding at assuming they are perfectly in phase.