A steel wire having a mass of 5g and length of 1.4m is fixed at both ends and has a tension of 968 N. (a) Find the speed of the transverse waves on the wire. (b) Find the frequency and the wavelength of the fundamental mode of vibration. (c) Find the frequencies of the second and third harmonics.
To answer these questions, we need to use the formulas related to waves on a stretched string:
1. Speed of transverse waves:
The formula to find the speed of transverse waves on a stretched string is given by:
Speed (v) = √(Tension (T) / Linear density (μ))
where Tension (T) is given as 968 N and Linear density (μ) is equal to mass (m) divided by length (L).
(a) Find the speed of the transverse waves on the wire:
Given: mass (m) = 5g = 0.005 kg, length (L) = 1.4m, Tension (T) = 968 N
So, Linear density (μ) = m / L = 0.005 / 1.4 = 0.0035714 kg/m
Speed (v) = √(968 / 0.0035714) ≈ 166.034 m/s
2. Frequency and wavelength of the fundamental mode of vibration:
The fundamental frequency of vibration is given by:
Fundamental frequency (f) = v / 2L
The wavelength (λ) of the fundamental mode of vibration is given by:
Wavelength (λ) = 2L
(b) Find the frequency and the wavelength of the fundamental mode of vibration:
Given: Length (L) = 1.4m, Speed (v) = 166.034 m/s
Fundamental frequency (f) = 166.034 / (2 * 1.4) ≈ 59.298 Hz
Wavelength (λ) = 2 * 1.4 = 2.8 m
3. Frequencies of the second and third harmonics:
The frequencies of the harmonics (n) can be calculated using the formula:
Frequency (fn) = nf
where n is the harmonic number and f is the fundamental frequency.
(c) Find the frequencies of the second and third harmonics:
Given: Fundamental frequency (f) = 59.298 Hz
Frequency of the second harmonic (f2) = 2 * 59.298 = 118.596 Hz
Frequency of the third harmonic (f3) = 3 * 59.298 = 177.894 Hz
Therefore:
(a) The speed of the transverse waves on the wire is approximately 166.034 m/s.
(b) The frequency of the fundamental mode of vibration is approximately 59.298 Hz, and the wavelength is approximately 2.8 m.
(c) The frequency of the second harmonic is approximately 118.596 Hz, and the frequency of the third harmonic is approximately 177.894 Hz.