A string has 0.9 kg/m of linear density and is under 90 N of tension. What is the angular
frequency of a sinusoidal wave applied on the string if you know that the average power of the
wave is 30W and its amplitude is 0.18 m?

To find the angular frequency (\(\omega\)) of the sinusoidal wave on the string, we can use the relationship between power, tension, linear mass density, amplitude, and angular frequency. The average power (\(P\)) transmitted by a sinusoidal wave on a string is given by:

\[
P = \frac{1}{2} \mu \omega^2 A^2 v
\]

Where:
- \(\mu\) is the linear mass density of the string (\(\mu = 0.9 \, \text{kg/m}\)),
- \(\omega\) is the angular frequency (what we're solving for),
- \(A\) is the amplitude of the wave (\(A = 0.18 \, \text{m}\)),
- \(v\) is the wave speed,
- \(P\) is the average power (\(P = 30 \, \text{W}\)).

First, let's calculate the wave speed (\(v\)) on the string using the tension (\(T\)) and the linear mass density (\(\mu\)):

\[
v = \sqrt{\frac{T}{\mu}}
\]

Given:
- \(T = 90 \, \text{N}\),
- \(\mu = 0.9 \, \text{kg/m}\),

\[
v = \sqrt{\frac{90 \, \text{N}}{0.9 \, \text{kg/m}}} = \sqrt{100} = 10 \, \text{m/s}
\]

Now that we have the wave speed, we can solve for \(\omega\) using the power equation:

\[
P = \frac{1}{2} \mu \omega^2 A^2 v
\]

Substituting the known values:

\[
30 = \frac{1}{2} \times 0.9 \times \omega^2 \times (0.18)^2 \times 10
\]

Simplifying:

\[
30 = 0.9 \times \omega^2 \times 0.0324 \times 5
\]

\[
30 = 0.1458 \times \omega^2
\]

Solving for \(\omega^2\):

\[
\omega^2 = \frac{30}{0.1458} \approx 205.79
\]

\[
\omega = \sqrt{205.79} \approx 14.35 \, \text{rad/s}
\]

Thus, the angular frequency \(\omega\) is approximately **14.35 rad/s**.

Would you like more details on the calculations or explanations? Here are some related questions to deepen your understanding:

1. How does the tension in the string affect the wave speed?
2. What is the relationship between wave amplitude and the energy transmitted by the wave?
3. How would the angular frequency change if the amplitude were doubled?
4. What effect does the linear mass density have on the wave properties?
5. How is the power transmitted by a wave related to its frequency and amplitude?

**Tip:** Understanding how wave speed relates to tension and mass density is crucial for solving wave problems efficiently.

A string has 0.9 kg/m of linear density and is under 90 N of tension. What is the angular frequency of a sinusoidal wave applied on the string if you know that the average power of the wave is 30W and its amplitude is 0.18 m?

Solve for the angular frequency of a wave on a string with a linear density of 0.9 kg/m and 90 N tension. Given an average power of 30W and amplitude of 0.18m, use wave mechanics formulas to find the angular frequency. Suitable for undergraduate physics students.

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