Question

The length of a wire increases by 8mm when a weight of 5 kg is suspended from it. If other things remain the same but the radius of the wire is doubled, what will be the increases in its length?
a) 5mm
b) 2mm
c) 7mm
d) 9mm

Answer 1

Hint: Young's modulus before and after remains the same for the wire, use the formula for both cases and calculate the change in length.

Complete step by step answer:
Here in this question we have the length of wire increased by 8mm.
 Solid objects will be deform when equal loads are applied to them, if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity in which the object fails to do so and remains in it deformed state.
According to the question:
In both cases, before and after increasing the radius, have the same Young's modulus which the elasticity coefficient of length of solid matters.
Young's modulus is expressed as;
$Y = \dfrac{{FL}}{{AI}}$;
Where F$ = $ forces on the wire
weight is 5 kg and
L$ = $ initial length
A$ = $ cross sectional Area
on which force is exerted & I is changed in length.
Now, the first case we get;
$Y = \dfrac{{FL}}{{AI}}$;
$I = \dfrac{{FL}}{{AY}} = 8\;{\text{mm}}$
When we double the radius of the cross sectional area (A) increases by a factor of 4, Since $A{r^2}$ so here we get, area for the second case is four times the area of the first case, i.e. $A' = 4A$
So here is the increase in length.
$I' = \dfrac{{FL}}{{A'Y}} = \dfrac{{FL}}{{4AY}}$
Comparing the two expressions we can see length in the second case increases and is $\dfrac{1}{4}$ time the length increases with a smaller cross sectional area. That is $I' = \dfrac{1}{4} = \dfrac{{8\;{\text{mm}}}}{4}$
 $ = 2\;{\text{mm}}$
Therefore the answer is 2mm.

So, the correct answer is “Option B”.

Note: This question is related to the elastic properties of solid matter. Since all things except the radius of wire is changed. Therefore using the formula we can calculate change in length easily.