QuestionSinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density of $4.00 \times 10^{-2} \mathrm{~kg} / \mathrm{m}$ . The source can dellver a maximum power of 280 W , and the string is under a tension of 90 N . What is the highest frequency $f$ at which the source can operate?
Part 1 of 5 - Conceptualize Increasing the source frequency increases the power carried by the wave. A power of 280 W will possibly occur at a frequency on the order of 100 Hz .
Part 2 of 5 - Categorize We will use the expression for power carried by a wave on a string.
Part 3 of 5 - Analyze The wave speed is given by $\begin{array}{l} v=\sqrt{\frac{T}{\mu}} \\ \sqrt{\frac{\square \mathrm{~N}}{\square \mathrm{~m} / \mathrm{s}}} \\ =\square 0^{-2} \mathrm{~kg} / \mathrm{m} \end{array}$

Studdy Solution
Solve for the highest frequency. Substitute the given values into the expression for $f^2$ :
$f^2 = \frac{280}{2 \pi^2 (4.00 \times 10^{-2}) (47.43) (0.05)^2}$
Calculate $f^2$ :
$f^2 = \frac{280}{2 \pi^2 \times 0.04 \times 47.43 \times 0.0025}$
$f^2 = \frac{280}{0.023561944}$
$f^2 = 11879.63$
Calculate $f$ :
$f = \sqrt{11879.63}$
$f \approx 109 \, \mathrm{Hz}$
The highest frequency at which the source can operate is approximately:
$\boxed{109 \, \mathrm{Hz}}$

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