Q8: Two sinusoidal waves with the same amplitude of 9.00 mm and the same wavelength travel together along a string that is stretched along an x axis. Their resultant wave is shown twice in fig., as valley A travels in the negative direction of the x axis by distance d=56 cm in 8 ms. The tick marks along the axis are separated by 10cm, and height H is 8mm. Let the equation of one wave be of the form where ϕ1 =0 and you must choose the correct sign in front of ω, for the equation of the other wave, what are (a)ym (b) k (c)ω (d) ϕ2 and (e) the sign in front of ω?
a) ym = 9.00 mm
b) k = 2π/λ
c) ω = kv
d) ϕ2 = -(2π/λ)xd
e) The sign in front of ω should be negative.
To answer this question, we need to understand the properties and equations of sinusoidal waves. Let's go step by step to find the answers:
(a) To find the amplitude, we can use the given information that the height H is 8mm. The amplitude (A) of a sine wave is the maximum displacement from its equilibrium position. Therefore, A = H = 8 mm.
(b) The equation of a sinusoidal wave is given by y = A sin(kx - ωt + ϕ), where k represents the wave number. The wave number is defined as 2π divided by the wavelength (λ). In this case, the tick marks on the x-axis are separated by 10 cm, which implies that the wavelength (λ) is equal to 10 cm. Therefore, k = 2π/λ = 2π/(10 cm) (Notice that I converted cm to meters for consistency in SI units).
(c) To find ω (angular frequency), we use the formula ω = 2πf, where f is the frequency of the wave. The frequency of a wave is the reciprocal of the period (T), which is the time taken for one complete cycle of the wave. In this question, we are given that the valley A travels a distance of d = 56 cm in 8 ms. The distance traveled by the wave in one period is equal to the wavelength (λ). Therefore, the period T = 8 ms. Now, we can calculate the frequency using f = 1/T. Once we have the frequency, we can find ω using the formula mentioned earlier.
(d) The phase constant ϕ is given as ϕ1 = 0 for one wave. For the other wave, we need to find ϕ2, which represents the phase difference between the two waves.
(e) Lastly, we need to find the sign in front of ω. This sign determines the direction of the wave. If it is positive, the wave travels in the positive x-axis direction, and if it is negative, the wave travels in the negative x-axis direction. To determine this sign, we can use the given information that valley A travels in the negative direction of the x-axis by distance d = 56 cm in 8 ms. From this, we can deduce the direction of the wave and the sign in front of ω.
By following these steps and calculations, you should be able to find the values for (a) ym (amplitude), (b) k (wave number), (c) ω (angular frequency), (d) ϕ2 (phase constant), and (e) the sign in front of ω.
To determine the characteristics of the other wave, let's analyze the given information and use the equation:
y = A*sin(kx - ωt + ϕ)
where:
- y is the displacement of a particle on the string at position x and time t,
- A is the amplitude of the wave,
- k is the wave number (2π/λ),
- λ is the wavelength of the wave,
- ω is the angular frequency (2πf),
- t is the time,
- ϕ is the phase constant.
(a) From the given information, we have the amplitude A = 9.00 mm.
(b) Using the information that the tick marks along the axis are separated by 10 cm, we can find the wavelength:
λ = 10 cm = 0.1 m
(c) To find the angular frequency ω, we can use the formula:
v = λf
Since the given information doesn't provide the velocity, we can use the fact that the valley travels a distance d = 56 cm in 8 ms to estimate the velocity:
v = d/t = 56 cm / 8 ms = 700 cm/s = 7 m/s
Now, we can determine the angular frequency ω:
v = ω / k
7 m/s = ω / (2π/0.1 m)
7 m/s = ω * (0.1 m / (2π))
7 m/s = ω * 0.01592 m
Thus, ω = (7 m/s) / 0.01592 m ≈ 438.61 rad/s
(d) Since one wave's phase constant ϕ1 is given to be 0, we'll use this as a reference. The valley A travels a distance d = λ/2, so at that point, the phase must be π radians.
ϕ2 = π
(e) Now, we need to determine the sign in front of ω. Since the wave travels in the negative direction of the x-axis, the sign in front of ω for the other wave should be negative.
In summary:
(a) Amplitude ym = 9.00 mm
(b) Wave number k = 2π/λ = 2π/0.1 m = 20π rad/m
(c) Angular frequency ω ≈ 438.61 rad/s
(d) Phase constant ϕ2 = π
(e) Sign in front of ω: Negative.