Question

A string with a mass of \(30.0 \mathrm{~g}\) and a length of \(2.00 \mathrm{~m}\) is stretched under a tension of \(70.0 \mathrm{~N}\). How much power must be supplied to the string to generate a traveling wave that has a frequency of \(50.0 \mathrm{~Hz}\) and an amplitude of \(4.00 \mathrm{~cm} ?\)

Short answer

Answer: The power required is 20.76 W.

Step-by-step solution

Calculate Mass Density of the String

To calculate the mass density of the string (\(\mu\)), we need to divide the mass (\(m\)) by the length (\(L\)) of the string. \(\mu = \frac{m}{L} = \frac{30.0 \, \text{g}}{2.00 \, \text{m}} = 15.0 \, \frac{\text{g}}{\text{m}}\) Note that we will need to use the SI units for mass. Since \(1 \text{g} = 0.001 \text{kg}\), then \(\mu = 15.0 \times 0.001 \, \frac{\text{kg}}{\text{m}} = 0.015 \, \frac{\text{kg}}{\text{m}}\)

Determine the Wave Velocity

Now find the wave velocity (\(v\)) using the equation \(v = \sqrt{\frac{T}{\mu}}\), where \(T\) is the tension and \(\mu\) is the mass density. \(v = \sqrt{\frac{70.0 \, \text{N}}{0.015 \, \frac{\text{kg}}{\text{m}}}} = \sqrt{\frac{70.0}{0.015}} \, \frac{\text{m}}{\text{s}} = 68.4 \, \frac{\text{m}}{\text{s}}\)

Calculate the Angular Frequency

We are given the frequency \(f = 50.0 \, \text{Hz}\). Now, we can find the angular frequency (\(\omega\)) using the formula \(\omega = 2\pi f\). \(\omega = 2\pi (50.0 \, \text{Hz}) = 100\pi \, \text{rad/s}\)

Compute the Power Required

With all the necessary values, we can calculate the power (\(P\)) using the equation: \(P = \frac{1}{2} \, \mu \, v A^2 \omega\), where \(A\) is the amplitude of the wave. \(P = \frac{1}{2} (0.015 \, \frac{\text{kg}}{\text{m}})(68.4 \, \frac{\text{m}}{\text{s}})(0.040 \, \text{m})^2 (100\pi \, \text{rad/s}) = 20.76 \, \text{W}\) The power required to generate the traveling wave with the given properties is \(20.76 \, \text{W}\).