An open ended air column of length 35 cm resonates with a speaker sounding a 500 Hz note. If the air is vibrating in the fundamental mode, what temperature is the air?
To determine the temperature of the air, we need to use the formula for the speed of sound in air. The speed of sound depends on the temperature of the air and the properties of the medium through which it travels.
The formula for the speed of sound in air is:
v = √(γ * R * T)
where:
v is the speed of sound,
γ is the adiabatic index for air (approximately 1.4),
R is the gas constant for air (approximately 287 J/(kg·K)),
T is the temperature of the air in Kelvin.
In this case, we can assume that the air in the open ended air column is vibrating at the fundamental frequency given by the speaker, which is 500 Hz. The fundamental frequency in an open ended tube is determined by the length of the column.
The formula for the fundamental frequency of an open ended tube is:
f = (v/2L)
where:
f is the frequency,
v is the speed of sound,
L is the length of the tube.
Given that the length of the column is 35 cm (or 0.35 meters) and the fundamental frequency is 500 Hz, we can solve for the speed of sound in air using the above equation:
500 Hz = (v/2*0.35m)
Solving for v:
v = 500 Hz * 2 * 0.35m = 350 m/s
Now, we can substitute this value of the speed of sound into the first equation to find the temperature (T):
350 m/s = √(1.4 * 287 J/(kg·K) * T)
Solving for T:
T = (350 m/s)^2 / (1.4 * 287 J/(kg·K)) = 2,722 K
Therefore, the temperature of the air is approximately 2,722 Kelvin.