Two pipes of equal length are each open at one end. Each has a fundamental frequency of 715 Hz at 300 K. In one pipe the air temperature is increased to 313 K. If the two pipes are sounded together, what beat frequency results?
v =331sqrt(T/273)
v1=331sqrt(300/273) =347 m/s
v2=331sqrt(313/273) =354 m/s
Fundamental frequency λ=4L
f=v/ λ =>
f1/f2 =v2/v1 =>
f2=f1(v2/v1)=715(354/347)=729 Hz
f(beat) =f2-f1= 729-715 =14 Hz
Ah, the melodious world of pipes and beats. Well, if we have two pipes with equal lengths and fundamental frequencies of 715 Hz, and the temperature of one pipe is increased to 313 K, we can expect some jazzy beat action.
Now, first things first. The frequency of a pipe depends on its length and the speed of sound in air, which in turn depends on temperature. As the temperature increases, so does the speed of sound, my friend.
Since one pipe remains at 300 K and the other pipe is cranking up the heat to 313 K, the change in frequency will be due to the change in speed of sound between the two pipes.
Now, let's have some fun with numbers. Assuming a linear relationship between temperature and speed of sound, a 1 K increase can lead to approximately a 0.6% increase in speed of sound.
Therefore, if we calculate the difference in speed of sound between 300 K and 313 K, we'll end up with a thrilling beat frequency.
So, grab your imaginary calculator, my friend, and let's crunch some numbers to find out the beat frequency that will make your ears dance with joy!
To find the beat frequency resulting from sounding the two pipes together, we first need to determine the frequency of each pipe at the new temperature. The fundamental frequency of a pipe can be calculated using the formula:
f = (n * v) / 2L
Where:
f is the fundamental frequency
n is the harmonic number (for the fundamental frequency, n = 1)
v is the speed of sound
L is the length of the pipe
Given that the two pipes have equal lengths, we can ignore the length factor and focus on the change in temperature.
The speed of sound, v, is directly proportional to the square root of the temperature, according to the formula:
v1 / v2 = sqrt(T1 / T2)
Where:
v1 and v2 are the speeds of sound at temperatures T1 and T2, respectively.
Let's calculate the new speed of sound for each pipe at 313 K:
v1 = sqrt(T1 / T2) * v2
v1 = sqrt(300 K / 313 K) * v2
Now we can calculate the new frequencies for each pipe:
f1 = (n * v1) / 2L
f2 = (n * v2) / 2L
Since the two pipes are sounded together, the beat frequency is the absolute difference between their frequencies:
beat frequency = |f1 - f2|
Substituting the values, we can now calculate the beat frequency.
To find the beat frequency resulting from sounding two pipes together, we need to first understand what beat frequency is.
The beat frequency is the difference between the frequencies of two interfering sound waves. When two waves of slightly different frequencies overlap, they create a beat, resulting in a periodic increase and decrease in the intensity of sound.
In this case, we have two pipes of equal length, each open at one end. The fundamental frequency of each pipe at 300 K is 715 Hz. We are asked to find the beat frequency when the air temperature in one pipe is increased to 313 K.
To solve this problem, we need to use the formula for the speed of sound in a gas:
v = √(γ * R * T),
where:
v is the speed of sound,
γ is the adiabatic constant,
R is the specific gas constant, and
T is the temperature.
In this case, the pipes are open at one end, and since they have the same length, the fundamental frequency is given by:
f = (v/2L),
where:
f is the frequency, and
L is the length of the pipe.
Now, let's find the speed of sound in the gas at 300 K and 313 K:
v1 = √(γ * R * T1),
v2 = √(γ * R * T2),
where T1 = 300 K and T2 = 313 K.
Next, we can calculate the fundamental frequency for both pipes at each temperature:
f1 = (v1/2L),
f2 = (v2/2L).
Finally, we find the beat frequency:
Beat frequency = | f1 - f2 |.
Substituting the values into the equations and performing the calculations will give us the beat frequency resulting from sounding the two pipes together.