Question

A \(3.00-\mathrm{m}\) -long string, fixed at both ends, has a mass of \(6.00 \mathrm{~g}\). If you want to set up a standing wave in this string having a frequency of \(300 . \mathrm{Hz}\) and three antinodes, what tension should you put the string under?

Short answer

To find the tension required in the string, follow these steps: 1. Calculate the linear mass density: μ = 0.00200 kg/m. 2. Calculate the period: T = 0.00333 s. 3. Determine the wavelength: λ = 2.00 m. 4. Calculate the wave velocity: v = 600 m/s. 5. Calculate the tension: T = 720 N. The tension required is 720 N.

Step-by-step solution

Determine the linear mass density and period

First, we'll calculate the linear mass density \(μ\) by dividing the mass of the string by its length: $$ μ = \frac{m}{L} = \frac{6.00\,\text{g}}{3.00\,\text{m}} = 2.00\,\text{g/m} = 0.00200\,\text{kg/m} $$ Next, we'll find the period \(T\) using the given frequency \(f\): $$ T = \frac{1}{f} = \frac{1}{300\,\text{Hz}} = 0.00333\,\text{s} $$

Find the wavelength

Since we have 3 antinodes in the string, it means we have \(1.5\) complete wavelengths over the length of the string. Therefore, we can find the wavelength \(λ\) by dividing the length of the string by \(1.5\): $$ λ = \frac{L}{1.5} = \frac{3.00\,\text{m}}{1.5} = 2.00\,\text{m} $$

Calculate the wave velocity

Now we have the wavelength \(λ\) and the period \(T\), we can find the wave velocity \(v\) using the wave equation: $$ v = \frac{λ}{T} = \frac{2.00\,\text{m}}{0.00333\,\text{s}} = 600\,\text{m/s} $$

Calculate the tension

Finally, we can use the formula \(v = \sqrt{\frac{T}{μ}}\) to determine the required tension in the string to create the desired standing wave: $$ T = μv^2 = (0.00200\,\text{kg/m})(600\,\text{m/s})^2 = 720\,\text{N} $$ The tension required to create the desired standing wave with a frequency of \(300\,\text{Hz}\) and three antinodes in the string is \(720\,\text{N}\).