Question

The load versus elongation graph for four wires of the same materials is shown in the figure. The thinnest wire is represented by the line:

(a) OC
(b) OD
(c) OA
(d) OB

Answer 1

Hint: To solve this question one must know about the concept of Young's Modulus. The relationship between stress (force per unit area) and strain is described by Young's modulus (proportional deformation in an object). Thomas Young, a British scientist, is honoured with the name of the Young's modulus. When a certain load is applied to a solid object, it deforms. If the thing is elastic, the body returns to its original shape after the pressure is released. Beyond a small amount of deformation, many materials are not linearly elastic. Young's modulus is a constant that only applies to materials that are linearly elastic.

Formula used:
 \[Y=\dfrac{FL}{A\Delta L}\], where Y is the young’s modulus of the wire, F is the force or the load on the wire. $\Delta L$is the elongation in wire and L is the original length and A is the cross-section area.

Complete answer:
Young's modulus is essentially a constant that varies depending on the substance. The Young's modulus gauges a material's resistance to length or size changes when an object composed of that material is compressed or stretched. It is sometimes referred to as the elasticity modulus.

We’ll make use of this concept here by using the formula:
\[Y=\dfrac{FL}{A\Delta L}\]

Let’s take the slope of the graph to be m. If we look at the graph then the slope of the graph will be $\dfrac{load}{elongation}$ or we can say that $m=\dfrac{Force}{Elongation}=\dfrac{F}{\Delta L}$.

Now using the formula \[Y=\dfrac{FL}{A\Delta L}\], we’ll find the value of the slope of the graph. If we see,
$\dfrac{F}{\Delta L}=\dfrac{YA}{L}$…… (i)
\[\therefore A=\pi {{r}^{2}}\]
Where, r is the radius of the wire.

We know that radius shows the thickness of the wire and higher the thickness, higher the radius. Now from equation (i) it’s visible that slope is directly proportional to the area of the wire. So, we need to find the graph with highest slope and we can see that the wire OD has the highest slope and thereby it will have the maximum thickness.

Hence, the correct option is B. OD

Note: Strength and resistance to deformation are two characteristics that are elaborated by the mechanical properties of solids. Strength is an object's capacity to endure applied tension, or how much stress it can withstand. How resistant an object is to changing shape is referred to as resistance to deformation. The object can easily change its shape if the resistance to deformation is low, and the opposite is also true.