A rope of diameter 1 cm breaks when the tension applied exceeds 500...

Breaking stress is a property of the material and remains constant for similar ropes.

Breaking stress is given by the ratio of breaking tension to the cross-sectional area:

$S = \dfrac{T}{A} = \dfrac{4T}{\pi d^2}$

Since $S$ is constant, the maximum tension $T$ is directly proportional to the square of the diameter $d$:

$T \propto d^2$

Using the given values, we can write:

$\dfrac{T_2}{T_1} = \left(\dfrac{d_2}{d_1}\right)^2$

Substituting $T_1 = 500 \text{ N}$, $d_1 = 1 \text{ cm}$, and $d_2 = 2 \text{ cm}$:

$\dfrac{T_2}{500} = \left(\dfrac{2}{1}\right)^2$

$\dfrac{T_2}{500} = 4$

$T_2 = 2000 \text{ N}$