A rope of length 2.00 m and mass 80 g is fixed at one end and tied...

The rope is fixed at one end and tied to a light string at the other end. The fixed end acts as a node and the end tied to the light string acts as a free end (antinode).

For a string fixed at one end and free at the other, the length $L$ is related to the wavelength $\lambda$ by:

$L = (2n + 1)\dfrac{\lambda}{4}$ for $n = 0, 1, 2, \dots$

The allowed wavelengths are given by:

$\lambda = \dfrac{4L}{2n + 1}$

Given $L = 2.00 \text{ m}$.

For the fundamental mode ($n = 0$):

$\lambda_0 = \dfrac{4 \times 2.00}{1} = 8.00 \text{ m}$

For the first overtone ($n = 1$):

$\lambda_1 = \dfrac{4 \times 2.00}{3} = \dfrac{8.00}{3} \approx 2.67 \text{ m}$

For the second overtone ($n = 2$):

$\lambda_2 = \dfrac{4 \times 2.00}{5} = \dfrac{8.00}{5} = 1.60 \text{ m}$