A string is tied across the opening of a deep well, at a height 5 m above the water level. The string has length 7.51m and linear density 4.17 g/m; any time there is a wind, the string vibrates like mad. The speed of sound in air is 343 m/s. Assuming that the resonance causing this vibration involves the lowest harmonics of both the string and the well, calculate the string's tension. Answer in units of N.
Match the fundamental lowest frequency of the string (string wave speed divided by twice the string length) to that of the well acoustic fundamental (treated as an open pipe) and solve for the tension, which will be the only unknown.
The well is a quarter wavelength long, compute frequency.
THen, knowing freq, mass density you can find tension from the law of strings.
A half wavelength actually, node both ends, antinode in the middle.
Oh, yes, the well is a quarter wave.
To elaborate on Prof Damons last remark: The water at the bottom provides a 180degree phase change, so the path up and down have to add to the other 180 degrees needed for wave reinforcement, thus 1/4 lambda length down, and 1/4 lambda up.
To calculate the string's tension, we can use the principle of resonance. Resonance occurs when the natural frequency of an object matches the frequency of an external force, causing the object to vibrate with maximum amplitude.
In this case, the resonance is between the string and the well. The length of the string is given as 7.51 m, which represents half of the wavelength of the lowest harmonic. The full wavelength can be calculated by multiplying the length by 2, giving us 15.02 m.
The speed of sound in air is given as 343 m/s, which represents the speed of the wave traveling down the well. The frequency of the wave is the speed divided by the wavelength, giving us:
Frequency = Speed / Wavelength = 343 m/s / 15.02 m = 22.86 Hz
Now, let's calculate the linear mass density of the string. Linear mass density is defined as the mass per unit length. In this case, it is given as 4.17 g/m. To convert grams to kilograms, divide by 1000:
Linear Mass Density = 4.17 g/m = 0.00417 kg/m
Next, we need to find the mass of the string. The mass can be calculated by multiplying the length of the string by the linear mass density:
Mass = Linear Mass Density * Length = 0.00417 kg/m * 7.51 m = 0.03128 kg
Now, we can use the formula for the tension in a string under resonance:
Tension = (Frequency²) * Mass * (Wavelength / 2)²
Plugging in the values we calculated:
Tension = (22.86 Hz)² * 0.03128 kg * (15.02 m / 2)²
Tension = 5251.189824 N
So, the tension in the string is approximately 5251.19 N.