In Bell's spaceship paradox, length contraction does indeed apply to the spaceships and the rope connecting them. The paradox involves two spaceships connected by a rope that are initially at rest relative to each other. The ships then accelerate away from each other, and according to the theory of special relativity, they experience length contraction in their direction of motion.
From the perspective of an observer on one of the spaceships, they would perceive the other spaceship and the rope to be contracted in length along the direction of their relative motion. This means that the rope connecting the spaceships would appear shorter than its original length in the observer's frame of reference.
As for the string breaking, it depends on the specific conditions and forces involved in the scenario. If the rope is not strong enough to withstand the tension caused by the acceleration or if the acceleration is too rapid, it is possible that the rope could break. However, the breaking of the rope in Bell's spaceship paradox is not directly related to length contraction itself.
A spacetime diagram can help illustrate the situation. In the rest frame of one spaceship, the diagram would show the other spaceship moving away at an angle, with the rope connecting them appearing contracted in length. The diagram can provide a visual representation of the relative positions and motions of the spaceships, but it does not inherently explain the breaking of the rope. The breaking of the rope would require considering the specific forces and tensions involved, which are beyond the scope of the spacetime diagram alone.