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 "-")?

To find the values for ym, k, ω, φ2, and the sign in front of ω, we need to analyze the information given in the question and use the properties of sinusoidal waves.

(a) Finding ym (amplitude):
The amplitude of both waves is given as 9.11 mm.

Therefore, ym = 9.11 mm.

(b) Finding k (wave number):
The wave number (k) can be determined using the formula:

k = (2π / λ)

where λ is the wavelength.

The wavelength is not given directly, but we can observe the distance between two adjacent tick marks on the x-axis, which is given as 12 cm.

Since one full wavelength consists of one peak and one valley, the distance between two adjacent tick marks represents half a wavelength.

Therefore, λ = 2 * 12 cm = 24 cm.

However, we need to convert this to meters for consistency:

λ = 24 cm * (1 m / 100 cm) = 0.24 m.

Now, we can calculate the wave number:

k = (2π / 0.24 m)

(calculate k)

Therefore, k ≈ 26.18 m⁻¹ (rounded to two decimal places).

(c) Finding ω (angular frequency) and the sign in front of ω:
The angular frequency (ω) can be determined using the formula:

ω = 2πf

where f is the frequency.

Frequency cannot be directly determined from the given information. However, we can calculate it using the distance traveled by valley A and the time it took.

The distance traveled by valley A (d) is given as 60.0 cm.

Since one full wavelength consists of one peak and one valley, the distance traveled by valley A represents half a wavelength.

Therefore, d = (1/2) λ.

Substituting the value of λ, we can solve for d:

60.0 cm = (1/2) * 24 cm = 12 cm.

Now, we can calculate the time period (T) of the wave using the formula:

T = (d / v)

where v is the wave velocity.

Wave velocity is not given directly, but we know that velocity is equal to (wavelength × frequency):

v = λf.

Using the given information, we have:

v = 0.24 m * f.

Now, we can solve for the time period (T):

T = (12 cm / (0.24 m * f)).

Given that time (t) is 8.0 ms (milliseconds), we can equate the time period (T) to t and solve for the frequency (f):

T = t = 8.0 ms = 8.0 x 10^(-3) s.

Now, we have:

8.0 x 10^(-3) s = (12 cm / (0.24 m * f)).

Solving for f:

f = (12 cm / (0.24 m * 8.0 x 10^(-3) s)).

Now, we can determine the angular frequency (ω):

ω = 2πf.

(calculate ω)

(round ω to two decimal places)

Therefore, ω ≈ 261.80 rad/s (rounded to two decimal places).

Lastly, we need to determine the sign in front of ω. The question states that valley A travels in the negative direction of the x-axis. Since wave motion is correlated to the sign of the angular frequency, the sign in front of ω should be "-".

(d) Finding φ2 (phase constant):
The question states that φ1 (phase constant for one wave) is 0. However, it does not explicitly mention the value of φ2 for the other wave.

Given that the resultant wave shown has a valley moving, we can infer that φ2 should be π radians (180 degrees).

Therefore, φ2 = π.

In summary:

(a) ym = 9.11 mm
(b) k ≈ 26.18 m⁻¹
(c) ω ≈ 261.80 rad/s
(d) φ2 = π
(e) The sign in front of ω is "-"