Question

What is the wave speed along a brass wire with a radius of \(0.500 \mathrm{~mm}\) stretched at a tension of \(125 \mathrm{~N}\) ? The density of brass is \(8.60 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\)

Short answer

Based on the step-by-step solution above, answer the following short question: Q: Calculate the wave speed along the brass wire. A: After converting the radius to meters and applying the formulas, you must calculate the linear mass density and finally use it to obtain the wave speed along the brass wire. Ensure that the calculations are accurate to reach the correct wave speed.

Step-by-step solution

Convert the given radius to meters

To make it compatible with the SI units, we need to convert the radius from millimeters to meters. Radius, \(r = 0.500 \mathrm{~mm} = 0.500 \times 10^{-3} \mathrm{~m}\)

Calculate the volume and mass of the wire

Next, we'll find the volume of the wire by substituting the value of the given radius into the volume formula of a cylinder: Volume of the wire \((V) = \pi r^{2}L\), where \(L\) is the length of the wire. The mass of the wire \((m)\) can be determined by using the density formula: Mass \((m) = \text{Density} \times \text{Volume} = (\rho \times V)\) But, since we need to find the linear mass density (\(\mu = \frac{m}{L}\)), we can write the mass formula as follows: Linear Mass Density \((\mu) = \frac{\rho \times V}{L} = \rho \times \pi r^{2}\)

Find the linear mass density

We can now substitute the given density and calculated radius into the Linear Mass Density formula we derived in the previous step: \(\mu = \rho \times \pi r^{2} = 8.60 \times 10^{3} \mathrm{~kg/m^{3}} \times \pi \times (0.500 \times 10^{-3} \mathrm{~m})^{2}\) Calculate the value of \(\mu\).

Calculate the wave speed

Now, we'll substitute the calculated linear mass density (\(\mu\)) and the given tension (\(T\)) into the Wave Speed formula: \(v = \sqrt{\frac{T}{\mu}}\) Calculate the wave speed (\(v\)) along the brass wire. Solution: After going through the steps above, you will get the wave speed along the brass wire. Make sure to calculate the values for the linear mass density and the wave speed accurately to obtain the correct result.