Question

A wire breaks when subjected to a stress S. If ρ is the density of the material of the wire and g, the acceleration due to gravity, then the length of the wire so that it breaks by its own weight is :
A ρgs
B ρ/gs
C gs/ρ
D s/ρg

Answer 1

To solve this question, we will apply Hooke’s law. Hooke’s law gives the ratio of stress to the strain and by applying Hooke’s law we can determine the length of the wire which is further related to strain.

Complete step by step answer:
We know that the weight of the wire is given by:
$W = \rho \cdot A \cdot l \cdot g$
Here ρ is the density of the material, A is the cross-sectional area of the wire, l is the length of the wire and g is the acceleration due to gravity, W is the weight of the wire.
We know that stress S is given by:
$\begin{array}{l} S = \dfrac{W}{A}\\\ = \dfrac{{\rho A \times l \times g}}{A}\\\ l = \dfrac{S}{{\rho g}} \end{array}$

Therefore, the correct option is (D).

Additional Information: Elasticity is a property of the material of a body due to which the body opposes any change in its shape or size when deforming forces are applied to it. However, they show this property up to a certain value of deforming force, beyond which they are unable to recover to its original state. Hence, the property of elasticity is completely lost, and the body is permanently deformed. The internal restoring force acting per unit area of the deformed body is called stress.

Note: In certain cases, it is taken that the weight acts at the center of gravity. As such for a wire, the lower half pulls the upper half. Under this condition the length required to break as calculated above doubles.