A string fixed at both ends is vibrating in its third harmonic. The...

The given equation of the standing wave is $y = 0.4 \sin(0.314 x) \cos(600\pi t)$.

Comparing this with the standard equation of a standing wave $y = A \sin(kx) \cos(\omega t)$, we get:

$k = 0.314 \text{ cm}^{-1}$

We know that the wave number $k = \dfrac{2\pi}{\lambda}$.

$\dfrac{2\pi}{\lambda} = 0.314$

Using $\pi \approx 3.14$, we have:

$\dfrac{2 \times 3.14}{\lambda} = 0.314$

$\lambda = \dfrac{6.28}{0.314} = 20 \text{ cm}$

For a string fixed at both ends vibrating in its third harmonic, the length of the string $L$ is given by:

$L = \dfrac{3\lambda}{2}$

Substituting the value of $\lambda$:

$L = \dfrac{3 \times 20}{2} = 30 \text{ cm}$