Question

105. A cable with a linear density of μ=0.2kg/m is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The speed of sound at the current temperature T=20°C is 343.00m/s . What are the frequency and wavelength of the hum?

Answer 1

The wavelength of the hum produced by the cable is approximately 4.44 meters, and the frequency is approximately 11.26 Hz, calculated using the known linear density, tension, and distance between poles.

Step-by-step explanation:

The student is asking about the properties of a standing wave created when the wind blows across a cable hung between two poles. Given the linear density (µ) of 0.2 kg/m, the tension in the cable (500 N), and the distance between poles (20 meters), we are to find the frequency and wavelength of the hum produced by the cable.

Firstly, we can calculate the wavelength (λ) of the wave using the information that 4.5 wavelengths fit between the two poles that are 20 meters apart:

λ = 20 meters / 4.5 wavelengths = 4.44 meters (approx)

Next, we can find the speed (v) of the wave on the cable using the formula:

v = sqrt(T / µ)

Plugging in the given values, we get:

v = sqrt(500 N / 0.2 kg/m) = sqrt(2500 m2/s2)

v = 50 m/s

Now, using the wave speed (v) and the wavelength (λ), we can calculate the frequency (f) of the wave using the formula:

f = v / λ

f = 50 m/s / 4.44 m = 11.26 Hz (approx)

Therefore, the wavelength is approximately 4.44 meters, and the frequency of the hum is approximately 11.26 Hz.

Answer 2

Answer: 11.5 Hz, 29.83 m

Step-by-step explanation:

Given

Linear density of the cable, μ = 0.2 kg/m

Tensión in the cables, T = 500 N

Wavelength of the wave, = 4.5 Waves

Distance between the poles, L = 20 m

Temperature of, t = 20° C

Speed of sound, v = 343 m/s

λ = length / number of waves =

λ = 20 / 4.5

λ = 4.44 m

Frequency of a standing wave is the same as frequency of a hum. Calculated using the formula

F = n/2L * √(T/μ)

F = 1/λ * √(T/μ)

F = 1/4.44 * √(500/0.2)

F = 0.23 * √2500

F = 0.23 * 50

F = 11.5 Hz

Wavelength of the hum,

λ = v/f

λ = 343 / 11.5

λ = 29.83 m