The length of an elastic string is a metres when the longitudinal tension is 4 N and b metres when the longitudinal tension is 5 N. The length of the string in metres when the longitudinal tension is 9 N is

The correct option is 5b-4a.

According to Hooke's Law, the extension of an elastic string is directly proportional to the tension applied to it. Let the natural length of the elastic string be $L_0$ meters, and let $k$ be the stiffness constant of the string. The relationship between the length $L$ of the string and the tension $T$ is given by:

$T=k(L-L_0)$

This can be rewritten to express the length $L$ as:

$L=L_0+\frac{T}{k}$

Let $c=\frac{1}{k}$ represent the compliance constant. The equation simplifies to:

$L=L_0+cT$

We are given the following conditions:

1. When the tension $T=4$ N, the length of the string is $a$ meters:

$a=L_0+4c$ --- (Equation 1)

2. When the tension $T=5$ N, the length of the string is $b$ meters:

$b=L_0+5c$ --- (Equation 2)

We need to find the length $L_9$ of the string when the tension is $9$ N:

$L_9=L_0+9c$ --- (Equation 3)

To solve for the unknowns, subtract Equation 1 from Equation 2:

$b-a=(L_0+5c)-(L_0+4c)$
$c=b-a$

Substitute this value of $c$ back into Equation 1 to find the natural length $L_0$:

$a=L_0+4(b-a)$
$a=L_0+4b-4a$
$L_0=5a-4b$

Now, substitute the expressions for $L_0$ and $c$ into Equation 3 to calculate $L_9$ under 9 N of tension:

$L_9=(5a-4b)+9(b-a)$
$L_9=5a-4b+9b-9a$
$L_9=5b-4a$

Thus, the length of the string when the longitudinal tension is 9 N is $5b-4a$ metres.