Two sinusoidal waves with the same amplitude of 9.11 mm and the same wavelength travel together along a string that is stretched along an x axis. Their resultant wave is shown twice in the figure, as valley A travels in the negative direction of the x axis by distance d = 60.0 cm in 8.0 ms. The tick marks along the axis are separated by 12 cm, and height H is 8.4 mm. Let the equation for one wave be of the form y(x, t) = ym sin(kx ± ωt + φ1), where φ1 = 0 and you must choose the correct sign in front of ω. For the equation for the other wave, what are (a) ym (in mm), (b) k, (c) ω, (d) φ2, and (e) the sign in front of ω (1 depicts "+" and 0 depicts "-")?
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To find the values of ym, k, ω, φ2, and the sign in front of ω, we need to analyze the given information and use the properties of sinusoidal waves.
First, let's understand the given information:
- The amplitude of the waves, ym, is 9.11 mm.
- The wavelength of the waves is not explicitly given, but we can figure it out from the distance between tick marks on the x-axis. Each tick mark represents 12 cm, which is equivalent to 0.12 m.
- The wave travels distance d = 60.0 cm in time t = 8.0 ms. This information allows us to find the speed of the wave, which is given by v = d/t.
To find the values of k and ω, we need to use the wave equation:
v = λf
where v is the speed of the wave, λ is the wavelength, and f is the frequency of the wave. Since both waves have the same wavelength and are traveling together, their frequencies are also the same.
From the equation above, we can rearrange to solve for f:
f = v/λ
Substituting the given values, we have:
v = d/t = 0.6 m/0.008 s = 75 m/s
λ = 0.12 m (from the tick mark separation)
f = 75/0.12 = 625 Hz
Now, we can use the relation between the angular frequency ω and the frequency f:
ω = 2πf
Substituting the value of f, we have:
ω = 2π * 625 ≈ 3925.2 rad/s
Next, we need to determine the phase shift φ2. Since the two waves are traveling together, the phase shift is related to the distance traveled, d.
The phase shift is given by:
φ2 = (2π/λ) * d
Substituting the values, we have:
φ2 = (2π/0.12) * 0.6 = π rad
Finally, we need to determine the sign in front of ω. The given equation has the form:
y(x, t) = ym sin(kx ± ωt + φ1)
From the given information, φ1 = 0, implying no phase shift in the first wave. We need to look at the figure to determine whether the second wave is ahead or behind the first wave.
If valley A of the second wave is traveling in the negative direction of the x-axis, this means the second wave is slightly ahead of the first wave. Therefore, the sign in front of ω is "+" (1).
To summarize:
(a) ym = 9.11 mm
(b) k = 2π/λ = 2π/0.12 ≈ 52.4 rad/m
(c) ω = 2πf ≈ 3925.2 rad/s
(d) φ2 = π rad
(e) The sign in front of ω is "+" (1).