The area of cross section of the rope used to lift a load by a crane is 2.5 × 10⁻⁴ m². The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be :
(take g=10 m s⁻² )

Solution

Physics sir · 3 steps

The rope breaks at a fixed stress, so area must grow in the same ratio as the load.

1. What stays fixed. The rope material can take only a certain maximum stress before it fails. Stress is force per unit area, σ = F/A, so the safe F/A is the same for the old rope and the new rope.
2. Maximum stress. For 10 metric tons, F = 10⁴ kg × 10 m s⁻² = 10⁵ N. So σₘₐₓ = 10⁵/(2.5 × 10⁻⁴) = 4 × 10⁸ N m⁻².

   F₁/A₁ = F₂/A₂ ⇒ A₂ = A₁ × F₂/F₁ = 2.5 × 10⁻⁴ × 25/10

3. So. A₂ = 6.25 × 10⁻⁴ m², which is option (a). Check: (2.5 × 10⁵)/(6.25 × 10⁻⁴) = 4 × 10⁸ N m⁻², the same maximum stress.

The idea to keep. Breaking stress is a property of the material, not the rope size. Same material means the same F/A, so a bigger load needs a proportionally bigger cross section.

Answer from the NTA final answer key. Solution by PrepWiser.