## Analyzing the Traveling Wave on the String:
**a) Wave Speed and Wavelength:**
1. **Tension and linear density:** We are given T = 20 N and ρ = 2.0 g/m.
2. **Wave speed:** v = √(T/ρ) = √(20 N / 2.0 g/m) ≈ 22.4 m/s.
3. **Frequency:** f = 100 Hz.
4. **Wavelength:** λ = v/f = 22.4 m/s / 100 Hz ≈ 0.224 m.
**b) Amplitude and Phase Constant:**
1. **Maximum displacement:** A = 1.0 mm = 0.001 m.
2. **Phase constant:** Φ is determined by the initial position of the hook. Since the hook is at its highest point at t = 0 s, Φ = 0.
**c) Displacement Equation:**
The general equation for a traveling wave is:
y(x, t) = A sin(kx - ωt + Φ)
where:
* k = 2π/λ is the wave number
* ω = 2πf is the angular frequency
Substituting the values:
y(x, t) = 0.001 sin(28.3 πx - 200πt + 0)
**d) Displacement at x = 0.50 m and t = 15 ms:**
Plugging in the values:
y(0.50 m, 0.015 s) ≈ 0.001 sin(14.15 π - 3.00 π) ≈ -0.0008 m (negative value indicates downward displacement)
**e) Re-calculating Phase Constant for Hook at Lowest Point:**
If the hook starts at its lowest point, it has a downward displacement at t = 0 s. This corresponds to a phase shift of π radians compared to the highest point case. Therefore, the new phase constant would be:
Φ = 0 + π = π
Substituting this new phase constant in the displacement equation would lead to a reversed cosine term, representing a wave starting with downward displacement at t = 0 s.
I hope this analysis provides a comprehensive understanding of the traveling wave on the string. Feel free to ask if you have any further questions about specific parts of the solution!
The probable question can be: A string with linear density 2.0 g/m is stretched along the positive x-axis with tension 20 N. One end of the string, at x = 0 m, is tied to a hook that oscillates up and down at a frequency of 100 Hz with a maximum displacement of 1.0 mm. At t = 0 s, the hook is at its highest point. (a) What are the wave speed on the string and the wavelength? (b) What are the amplitude and phase constant of the wave? (c) Write the equation for displacement y(x, t) of the traveling wave. (d) What is the string's displacement at x 0.50 m and t = 15 ms? (e) Redo the calculation of the phase constant for the case that the hook started (at t = 0 s) at its lowest point 4.