Answer :
To determine the frequency of traveling waves on the string, given the amplitude and average power, we need to use the formulas for average power and the relationship between frequency and wavelength.
The average power (P) transmitted by a wave on a string is given by the formula:
P = (1/2) * μ * ω^2 * A^2 * v
Where:
P is the average power
μ is the linear mass density of the string (mass per unit length)
ω is the angular frequency of the wave
A is the amplitude of the wave
v is the velocity of the wave
We need to find the frequency (f) of the waves, which is related to the velocity (v) and wavelength (λ) by the equation:
v = λ * f
Since the string is 3.81 m long, the wavelength is equal to the length of the string:
λ = 3.81 m
We can rearrange the equation to solve for f:
f = v / λ
Now, we can substitute the given values into the formulas. The linear mass density (μ) can be calculated by dividing the mass (m) by the length (L) of the string:
μ = m / L
μ = 171 g / 3.81 m
μ = 0.045 g/m = 0.045 kg/m
The tension in the string (T) is given as 39.1 N. The velocity (v) can be calculated using the formula:
v = √(T / μ)
v = √(39.1 N / 0.045 kg/m)
v ≈ 93.27 m/s
Next, we can substitute the values of μ, A, and v into the power formula to solve for ω:
P = (1/2) * μ * ω^2 * A^2 * v
Solving for ω:
ω^2 = (2P) / (μ * A^2 * v)
ω^2 = (2 * 78.3 W) / (0.045 kg/m * (5.70 mm)^2 * 93.27 m/s)
ω^2 ≈ 240.86 rad^2/s^2
Taking the square root of both sides:
ω ≈ 15.52 rad/s
Finally, we can calculate the frequency using the equation:
f = v / λ
f = 93.27 m/s / 3.81 m
f ≈ 24.46 Hz
Therefore, the frequency of the traveling waves on the string must be approximately 24.46 Hz for the average power to be 78.3 W.
To achieve an average power of 78.3 W, the frequency of the traveling waves on the string should be approximately 24.46 Hz, given a string length of 3.81 m, a mass of 171 g, and a tension of 39.1
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