A stretched string has a mass per unit length of 5.94 g/cm and a tension of 19.8 N. A sinusoidal wave on this string has an amplitude of 0.142 mm and a frequency of 159 Hz and is traveling in the negative direction of an x axis. If the wave equation is of the form y(x,t) = ym sin(kx + wt), what are (a) ym, (b) k, and (c) w, and (d) the correct choice of sign in front of w?
(a) – Exact number no tolerance, (b) and (c) – The tolerance is ± 1 in the 3rd significant digit
The second pic is the available units for the answers
The Correct Answer and Explanation is :
To determine the parameters of the sinusoidal wave on the stretched string, we will calculate the amplitude (ym), wave number (k), angular frequency (ω), and the correct sign in front of ω in the wave equation.
Given Data:
- Mass per unit length (μ) = 5.94 g/cm = 0.0594 kg/m
- Tension (T) = 19.8 N
- Amplitude (A) = 0.142 mm = 0.000142 m
- Frequency (f) = 159 Hz
(a) Amplitude (ym):
The amplitude of the wave is given as 0.142 mm, which is 0.000142 m. Therefore,
ym = 0.000142 m
(b) Wave Number (k):
The wave number k is related to the wavelength λ by the equation k = 2π/λ. To find λ, we use the relationship between wave speed (v), frequency (f), and wavelength:
v = f × λ
First, calculate the wave speed (v) using the formula:
v = √(T / μ)
Substituting the given values:
v = √(19.8 N / 0.0594 kg/m) ≈ 18.26 m/s
Now, calculate the wavelength (λ):
λ = v / f = 18.26 m/s / 159 Hz ≈ 0.114 m
Finally, calculate the wave number (k):
k = 2π / λ ≈ 2π / 0.114 m ≈ 55.1 rad/m
(c) Angular Frequency (ω):
The angular frequency ω is related to the frequency f by the equation ω = 2πf. Substituting the given frequency:
ω = 2π × 159 Hz ≈ 999.6 rad/s
(d) Correct Sign in Front of ω:
The wave is traveling in the negative direction of the x-axis, which corresponds to a wave function of the form y(x,t) = ym sin(kx + ωt). Therefore, the correct sign in front of ω is positive.
Summary of Results:
- Amplitude (ym): 0.000142 m
- Wave Number (k): 55.1 rad/m
- Angular Frequency (ω): 999.6 rad/s
- Correct Sign in Front of ω: Positive
These calculations are based on standard wave equations and the relationships between wave parameters.
For a visual explanation of how wave speed depends on tension and linear mass density, you might find the following video helpful:
videoWave Speed on a Stretched Stringturn0search6