Question

A steel wire of one meter long and one millimeter square area takes $200N$ force to stretch by one millimeter. What force is required to stretch the same wire from $10m$ to $1002cm$.
$\begin{aligned} & a)100N \\\ & b)200N \\\ & c)400N \\\ & d)2000N \\\ \end{aligned}$

Answer 1

Let us find the young’s modulus of the steel wire in the first case. As the material given is same and the area of cross section is also same, we can use the same young’s modulus for the second case where the change in length is given and the rest is given in the question.

Formula used: $Y=\dfrac{Fl}{A\Delta l}$

Complete step by step answer:
let us find the young’s modulus of the given steel wire first,
$\begin{aligned} & Y=\dfrac{Fl}{A\Delta l} \\\ & Y=\dfrac{200\times 1}{1\times 1} \\\ & \Rightarrow Y=200 \\\ \end{aligned}$
Change in length in second case is
$\begin{aligned} & \Delta l=1002-1000cm \\\ & \Rightarrow \Delta l=2cm \\\ \end{aligned}$
Now, we know the young’s modulus, area, length of the steel wire and the change in length, therefore, we can find the force acting easily.
$\begin{aligned} & 200=\dfrac{F\times 1}{1\times 2} \\\ & F=400N \\\ \end{aligned}$

So, the correct answer is “Option C”.

Additional Information: The young’s modulus or also called the modulus of elasticity in tension is a property of mechanical nature that measures the tensile stiffness of a solid material. It quantifies the relationship between tensile stress and the axial strain. This was named after the British scientist Thomas young. The term modulus is derived from the Latin root term meaning measure. A solid material will undergo elastic deformation when a small load is applied to it in compression or extension. Elastic deformation is reversible because we can get the normal shape back after removing the load.

Note: In the above equation, the change in length given in the second case is of the same material given. Therefore, the young’s modulus derived in the first case can be used in the second case too. The young’s modulus depends on the force acting, area of the material, length and change in length.