Question

When the tension in a cord is \(75 \mathrm{N}\), the wave speed is $140 \mathrm{m} / \mathrm{s} .$ What is the linear mass density of the cord?

Short answer

Answer: The linear mass density of the cord is approximately 0.0038 kg/m.

Step-by-step solution

Rearrange the formula for the linear mass density

In order to find the linear mass density (\(\mu\)), we'll first rearrange the formula \(v = \sqrt{\frac{T}{\mu}}\). Square both sides of the equation to get rid of the square root: \(v^2 = \frac{T}{\mu}\) Now, multiply both sides by \(\mu\): \(\mu v^2 = T\) Finally, divide both sides by \(v^2\) to solve for \(\mu\): \(\mu = \frac{T}{v^2}\)

Substitute the given values

Now that we have the formula \(\mu = \frac{T}{v^2}\), we can plug in the given values for tension (\(T=75\,\mathrm{N}\)) and wave speed (\(v=140\,\mathrm{m/s}\)): \(\mu = \frac{75\,\mathrm{N}}{(140\,\mathrm{m/s})^2}\)

Calculate the linear mass density

Finally, we calculate the value of the linear mass density: \(\mu = \frac{75\,\mathrm{N}}{19600\,\mathrm{m^2/s^2}}\) \(\mu = 0.0038265\,\mathrm{kg/m}\) So the linear mass density of the cord is approximately \(0.0038\,\mathrm{kg/m}\).