A string has linear density $10.0 \cdot 10^{-3} \, \mbox{kg/m}$ and is kept under a tension of 100 N. A sinusoidal transverse wave, with a wavelength of 0.30 m, is traveling in the positive direction of an $x$ axis. What is the frequency of the wave?
We are given $\mu = 10.0 \cdot 10^{-3} \, \mbox{kg/m}$, $ \tau = 100 \, \mbox{N}$ and $\lambda = 0.30 \, \mbox{m}$. Furthermore, we know that
$$v = f\cdot \lambda = \sqrt{\frac{\tau}{\mu}}.$$
Hence
$$\displaystyle f =
\frac{\sqrt{\tau / \mu}}{\lambda} =
\frac{\sqrt{(100 \, \frac{\mbox{kg}\cdot\mbox{m}}{\mbox{s}^2}) /
(10.0 \cdot 10^{-3} \frac{\mbox{kg}}{\mbox{m}})}}{0.30 \, \mbox{m}} =
\frac{\sqrt{100 \cdot 10^{3} / 10.0 \, \frac{\mbox{m}^2}{\mbox{s}^2}}}
{0.30 \, \mbox{m}} = 300 \, \mbox{Hz}.
$$
I know my calculations are right per se, however, I am unsure about the significant figures. The answer is "supposed" to be 333 Hz. But as we are given $N = 100$ – that corresponds to only one significant figure – isn't 333 Hz simply wrong?
Best Answer
You are both right and wrong.
$N=100$ is quote to 3 significant figures (i.e. $1.00 \times 10^{2}$). If this were to be quoted to 1 significant figure it would be $1 \times 10^{2}$.
Similarly, $10.0 \times 10^{-3}$ is given to 3 significant figures.
However $\lambda = 0.30 m$ is only given to two significant figures (i.e. $3.0 \times 10^{-1}$ m).
Finally, you have made a mistake in your calculation. The numerical answer is 333.33 Hz, but as wavelength is only given to 2 significant figures, it might be best to quote this as 330 Hz or $3.3\times 10^2$ Hz.