Chapter 16, Problem 16.1P

To determine

Verify that the quantity $c=\sqrt{T/\mu }$ that appears in the wave equation for a string does indeed have the units of speed.

Answer to Problem 16.1P

Verified that the quantity $c=\sqrt{T/\mu }$ that appears in the wave equation for a string does indeed have the units of speed.

Explanation of Solution

Given that, $c=\sqrt{T/\mu }$, where $c$ is the speed of light, $T$ is the tension, and $\mu$ is the linear mass density.

The unit of $c$ is $\text{m}/\text{s}$, the unit of $T$ is $\text{N}$, and the unit of $\mu$ is $\text{kgm}/\text{s}$.

Use $\text{m}/\text{s}$ for $c$, $N$ for $T$, and $\text{kgm}/\text{s}$ for $\mu$ in the equation $c=\sqrt{T/\mu }$.

    $\text{m}/\text{s}=\sqrt{\frac{\text{N}}{\text{kgm}/{\text{s}}^{\text{2}}}}$

But $1\text{N}=1\text{kgm}/{\text{s}}^{\text{2}}$. Use $1\text{kgm}/{\text{s}}^{\text{2}}$ for $1\text{N}$ in the above equation.

    $\begin{array}{c}\text{m}/\text{s}=\sqrt{\frac{1\text{kgm}/{\text{s}}^{\text{2}}}{\text{kgm}/\text{s}}}\\ =\sqrt{{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =\text{m}/\text{s}\end{array}$

Hence the given problem is verified.

Conclusion:

Therefore, the quantity $c=\sqrt{T/\mu }$ that appears in the wave equation for a string does indeed have the units of speed.