A harpsichord string of length 1.60 m and linear mass density 25.0 mg/m vibrates at a (fundamental) frequency of 450.0 Hz.

(a) What is the speed of the transverse string waves in m? (b) What is the tension in N? (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.
wavelength m
frequency Hz

To solve this problem, we will use the following formulas:

(a) Speed of transverse waves on a string:
v = √(T/μ)
where v is the speed of waves on the string, T is the tension in the string, and μ is the linear mass density of the string.

(b) Tension in the string:
T = (λ/2) * f² * μ
where λ is the wavelength of the sound wave, f is the frequency of the sound wave, and μ is the linear mass density of the string.

(c) Relationship between speed, wavelength, and frequency:
v = λ * f
where v is the speed of sound in air, λ is the wavelength of the sound wave, and f is the frequency of the sound wave.

Now, let's plug in the given values and solve for each part:

(a) Speed of transverse waves on the string:
v = √(T/μ)
Given:
μ = 25.0 mg/m = 25.0 x 10^-6 kg/m
v = √(T/μ) = √(T/(25.0 x 10^-6))
We need to find the tension (T) in the string to solve for v.

(b) Tension in the string:
T = (λ/2) * f² * μ
Given:
λ = length of the string = 1.60 m
f = frequency of the string = 450.0 Hz
μ = 25.0 x 10^-6 kg/m
T = (1.60/2) * (450.0)^2 * (25.0 x 10^-6)
Calculate T.

(c) Relationship between speed, wavelength, and frequency:
v = λ * f
Given:
v = 340 m/s (speed of sound in air)
λ = wavelength
f = frequency of the sound wave (same as the frequency of the string)
We can rearrange the equation to solve for λ:
λ = v / f
Plug in the values to find λ.

I hope this clarifies the steps needed to solve the problem.