Tranverse waves and velocity problem

[dmolson]

A piano string of length 1.6 m and mass density 26 mg/m vibrates at a (fundamental) frequency of 450 Hz.

(a) What is the speed of the transverse string waves?
(b) What is the tension?
c) What are the wavelength and frequency of the sound wave in air produced by vibration of the string? The speed of sound in air at room temperature is 340 m/s.

I have been working on this problem and cannot seem to come up with the solution. Any help would be greatly appreciated.

 

[Doc Al]

What have you done so far?

What's the wavelength of the fundamental mode? How are speed, wavelength, and frequency related?

What's the relationship between tension, mass density, and wave speed?

 

[kyle8921]

(a) The speed of transverse waves on a string can be calculated using the formula v = √(T/μ), where v is the speed, T is the tension, and μ is the mass density. In this case, the mass density is given as 26 mg/m, which is equivalent to 0.026 kg/m. The frequency of the string is given as 450 Hz, which is the same as 450 vibrations per second. Using the given length of 1.6 m, we can calculate the wavelength using the formula λ = v/f, where λ is the wavelength, v is the speed, and f is the frequency. Substituting in the values, we get λ = (0.026 kg/m/450 Hz) = 0.0000578 m. This means that the speed of the transverse waves on the string is v = √(T/μ) = √(T/0.026 kg/m) = √(T/0.026) = √(T/0.026) = √(T/0.026) = 0.00904 m/s.

(b) The tension of the string can be calculated using the formula T = μv², where T is the tension, μ is the mass density, and v is the speed. Substituting in the values, we get T = (0.026 kg/m)(0.00904 m/s)² = 0.0000218 N. This is the tension required to produce the given frequency of 450 Hz on the string.

(c) The wavelength of the sound wave produced by the vibrating string can be calculated using the formula λ = v/f, where λ is the wavelength, v is the speed of sound in air at room temperature (340 m/s), and f is the frequency of the string (450 Hz). Substituting in the values, we get λ = (340 m/s)/(450 Hz) = 0.7556 m. This is the wavelength of the sound wave produced by the vibrating string. The frequency of the sound wave will be the same as the frequency of the string, which is 450 Hz.