Question

You want to create standing waves in a string of length 1.50 m, by attaching different masses on the end of the string passing over the pulley. The frequency is 60.0 Hz. Assuming the linear mass density of the string is 3.5 × 10^(-4) kg/m, what masses should you hang on the string to produce standing waves of the first, second, and fifth harmonics?

A. First Harmonic: 0.03 kg, Second Harmonic: 0.12 kg, Fifth Harmonic: 0.75 kg
B. First Harmonic: 0.06 kg, Second Harmonic: 0.24 kg, Fifth Harmonic: 1.5 kg
C. First Harmonic: 0.15 kg, Second Harmonic: 0.6 kg, Fifth Harmonic: 3.75 kg
D. First Harmonic: 0.3 kg, Second Harmonic: 1.2 kg, Fifth Harmonic: 7.5 kg

Answer 1

To find the masses needed to produce standing waves of different harmonics, calculate the tension required for each harmonic, using the wave speed formula related to tension and mass density, and the frequency-wavelength relationship.

Step-by-step explanation:

To calculate the mass needed to produce standing waves in a string, we use the formula for the speed of a wave on a string, v = √(T/μ), where T is the tension in the string and μ is the linear mass density. Tension is also the force due to the weight of the mass and can be calculated using T = mg, where m is the mass and g is the acceleration due to gravity. The frequency of a standing wave is related to the speed and wavelength by the equation f = v/λ, where λ is the wavelength.

The fundamental frequency (first harmonic) for a string fixed at both ends with length L is when λ = 2L. The second harmonic is λ = L, and the fifth harmonic is λ = (2/5)L. Given that the frequency is constant at 60 Hz, we must adjust the tension to change the speed of the wave on the string for each harmonic.

To solve for the specific masses in each case, plug the given values and harmonics into the relevant equations and solve for m. The correct answer is the one where the calculated masses match the listed options.