Question

The length of an elastic string is x meter when the Tension is 8N and y meter when tension is 10N. The length in meter when the tension is 18N is:
(A) 4x-5y
(B) 5y-4x
(C) 9x-4y
(D) 4x-9y

Answer 1

When force is applied to an elastic material in the elastic limit of that material then its length/area/volume changes. On removal of force, the material tends to return to its original size.Here we are concerned with one-dimensional string so its length will increase.

Formula used:
By the mathematical definition of Young’s modulus we can solve the above problem given as:
$Y = \dfrac{{stress}}{{strain}} = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta l}}{l}}}$
Where,
F is the force applied
A is the area of cross section of the string
$\Delta l$ is the change in length
$l$ is the original length

Complete step by step answer:
Elasticity is the property by which the material regains its original shape and size when the deformative force is removed. This force should not exceed the elastic limit of the material otherwise the material will break and will not regain its original size and shape.According to the Hooke’s Law the stress applied is directly proportional to the strain. The proportionality constant which appears is the Young’s modulus in the case of one dimension.
So, in terms of equation it can be written as:

$$Y = \dfrac{{stress}}{{strain}} = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta l}}{l}}} \\\ \Delta l = F.\dfrac{l}{{YA}} \\\ \Delta l = cF \\\$$

Here, the original length, young’s modulus and the area of cross section are constants, so we have replaced it by c. According to the question,
In the 1st scenario:
$x - l = 8c$
In the 2nd scenario:
$y - l = 10c$
Now, dividing both equations to get the value of original length:

$$\dfrac{{x - l}}{{y - l}} = \dfrac{8}{{10}} \\\ \Rightarrow 5x - 5l = 4y - 4l \\\ \Rightarrow l = 5x - 4y \\\$$

Substituting the value of l in any one of the equations to find constant c:

$$x - 5x + 4y = 8c \\\ c = \dfrac{1}{2}(y - x) \\\$$

Now for the required scenario (substituting the value of l and c):

$${l_{req}} - l = 18c \Rightarrow {l_{req}} = 18c + l \\\ \Rightarrow 18.\dfrac{1}{2}(y - x) + 5x - 4y \\\ \Rightarrow {l_{req}} = 5y - 4x \\\$$

The correct answer is option B.

Note: Most of the students get confused with the parameter $\Delta l$. It is the change in length i.e. the increased length – original length. It is commonly misinterpreted as the new length when force is applied because of the notation $l^`$ used in many books.