Question:

A transverse wave is traveling along a string with a frequency of 50 Hz and a wavelength of 0.4 m. The amplitude of the wave is 0.2 m.

a) Calculate the wave speed.

b) Determine the period of the wave.

c) Find the maximum particle velocity and acceleration of the string.

d) If the wave is reflected back from the fixed end of the string, state whether there will be a phase change or no phase change in the reflected wave.

Answer:

a) The wave speed (v) of a wave can be calculated using the formula:

    v = f * λ,

where f is the frequency and λ is the wavelength.

Given that the frequency is 50 Hz and the wavelength is 0.4 m, we can substitute these values into the formula:

    v = 50 Hz * 0.4 m
    v = 20 m/s

Therefore, the wave speed is 20 m/s.

b) The period (T) of a wave is the reciprocal of its frequency, so we can calculate it using the formula:

    T = 1 / f

Given that the frequency is 50 Hz, we can substitute this value into the formula:

    T = 1 / 50 Hz
    T = 0.02 s

Therefore, the period of the wave is 0.02 seconds.

c) The maximum particle velocity (vₚₘₐₓ) of a transverse wave is given by the formula:

    vₚₘₐₓ = A * ω,

where A is the amplitude of the wave and ω is the angular frequency.

The angular frequency (ω) can be calculated using the formula:

    ω = 2π/T,

where T is the period of the wave.

Given that the amplitude is 0.2 m and the period is 0.02 s, we can substitute these values into the formulas:

    ω = 2π / 0.02 s
    ω = 314.16 rad/s

    vₚₘₐₓ = 0.2 m * 314.16 rad/s
    vₚₘₐₓ = 62.83 m/s

Therefore, the maximum particle velocity of the string is 62.83 m/s.

The maximum particle acceleration (aₚₘₐₓ) can be calculated using the formula:

    aₚₘₐₓ = A * ω²,

Given that the amplitude is 0.2 m and the angular frequency is 314.16 rad/s, we can substitute these values into the formula:

    aₚₘₐₓ = 0.2 m * (314.16 rad/s)²
    aₚₘₐₓ = 19630.86 m/s²

Therefore, the maximum particle acceleration of the string is 19630.86 m/s².

d) When a transverse wave is reflected from a fixed end, the wave undergoes a phase change of 180 degrees (or π radians). Therefore, there will be a phase change in the reflected wave.

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In summary, the answers to the questions are:

a) The wave speed is 20 m/s. b) The period of the wave is 0.02 seconds. c) The maximum particle velocity of the string is 62.83 m/s, and the maximum particle acceleration is 19630.86 m/s². d) There will be a phase change in the reflected wave.