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A uniform rope of length L and density ρ is suspended vertically from a fixed support. If the Young’s modulus of the rope material is Y, then the extension produced in the rope by its own weight is

1. (1/4) ρgL²/Y
2. (1/2) ρgL²/Y
3. (ρgL²)/Y
4. (ρgL²)/(2Y)

Correct answer: (ρgL²)/(2Y)

Solution

The correct option is derived from the relationship between the weight of the rope and its extension under its own weight, where the extension is proportional to the square of the length and inversely proportional to the Young's modulus. The factor of 1/2 arises from integrating the varying forces along the length of the rope.

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