Question

Two nails are hammered into the wall at the same height off the ground and with a horizontal separation of d=13 cm. A loop of string (mass density μ=4.0 g/m ) is hung around the nails so that the nails are inside the loop. Then a mass m=1.9 kg is hung from the bottom segment of string, and as a result the string forms a taut equilateral triangle. (Assume that the nails behave like frictionless pulleys, and we may model the tension as being the same everywhere in the loop of string.) If the top segment of string were lightly plucked, what fundamental frequency f would be heard? For the limit check, investigate what happens to f if the hanging mass grows large (m→[infinity])

Answer 1

To find the fundamental frequency of the string, we need to consider the wave properties of the string, including the wavelength, speed of the wave, and tension in the string. By calculating the tension and linear mass density, we can determine the frequency using the equation v = λf. In the case of a large hanging mass, the frequency would approach zero due to the increased tension in the string.

Step-by-step explanation:

To find the fundamental frequency of the string when the top segment is lightly plucked, we need to consider the wave properties of the string. The string forms an equilateral triangle, which means the wavelength is twice the distance between the nails. The wavelength and frequency are related by the formula v = λf, where v is the speed of the wave. In this case, the speed of the wave is determined by the tension in the string and the linear mass density. The frequency f can be calculated using the formula f = v / λ. Therefore, to calculate the frequency, we need to find the tension in the string and the linear mass density.

To find the tension in the string, we can consider the forces acting on the hanging mass. The tension in the bottom segment of the string is equal to the weight of the hanging mass, which is m * g, where m is the mass of the hanging mass and g is the acceleration due to gravity.

The linear mass density μ can be calculated by dividing the mass of the string by its length.

Substituting the values into the formula f = v / λ, we can calculate the frequency. As for the limit check when the hanging mass grows large, the frequency would approach zero because the tension in the string would increase significantly, making it harder for the string to vibrate.