Question

A spring, which is initially in its unstretched condition, is first stretched by a length $x$ and then again by a further length $x$. The work is done in the first case $W_1$ and in the second case $W_2$. Find $\frac{W_2}{W_1}$.

Answer 1

Given that

A spring, which is initially in its unstretched condition,

It is first strectched by $x$ m

We know that

Work done by stretching $X$ m from initial position is , $W=-\frac{1}{2}(X^2-x_0^2)$

where $x_0$ is the initial position

In First Case :

$X=x$ m
$x_0=0$
Hence $W_1=-\frac{1}{2}{x}^2$

In Second Case :

$X=2x$ m

$x_0=x$ m

Hence , $W_2=-\frac{3}{2}{x}^2$

So, Required Ratio, $\frac{W_2}{W_1}=3$