A \(500 \mathrm{~g}\) ball is attached to one end of an aluminum wire of area of cross-section \(0.5 \mathrm{~mm}^2\) and an unstretched length of \(1.4 \mathrm{~m}\). The other end of the wire is fixed to the top of a vertical pole. The ball rotates about the pole in a horizontal plane such that the angle between the wire and the horizontal is \(30^{\circ}\). The increase in the length of the wire is ____ \(\mathrm{mm}\). (Young's modulus of aluminum \(=0.7 \times 10^{11} \mathrm{Nm}^{-2}\) and acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\))

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.