Expert Answer
To calculate the natural frequency of the system, we need to determine the equivalent spring constant and mass associated with the given system. Here's how you can calculate it:
Determine the volume of the line: The volume of the line can be calculated using the formula for the volume of a cylinder:
Volume = ? * (diameter/2)^2 * length
Volume = ? * (6.0 mm / 2)^2 * 1 m = 0.0906 m^3
Determine the mass of the air in the line: The mass of the air can be calculated using the ideal gas law and assuming air behaves ideally:
PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
Rearranging the equation, we have n = PV / RT.
Assuming air is at atmospheric pressure (since the pressure given is above atmospheric pressure), we can use 1 atm as the pressure:
n = (9 bar * 1 atm / 1.01325 bar) * (0.0906 m^3) / (8.314 J/(mol·K) * (30 + 273.15) K)
n ? 0.00463 mol
The molar mass of air is approximately 28.97 g/mol, so the mass of air in the line is:
Mass = n * molar mass = 0.00463 mol * 28.97 g/mol = 0.134 g
Calculate the equivalent spring constant: The equivalent spring constant can be calculated using the formula:
k = (P * V) / ?V
Here, P is the pressure, V is the volume, and ?V is the change in volume due to compression. The change in volume is equal to the volume of the transducer, which is given as 15 cm^3 (or 0.015 L):
k = (9 bar * 1 atm / 1.01325 bar) * (0.0906 m^3) / (0.015 L)
k ? 53,250 N/m
Calculate the natural frequency: The natural frequency can be calculated using the formula:
f = (1 / 2?) * ?(k / m)
Here, k is the spring constant calculated in the previous step, and m is the mass calculated in step 2.
f = (1 / 2?) * ?(53,250 N/m / 0.134 kg)
f ? 21,273 Hz (rounded to the nearest whole number)