Question

An elastic spring of unstretched length $L$ and force constant $k$ is stretched by a small length $x .$ It is further stretched by another small length $y$. Work done during the second stretching is

• A. $\frac{k y}{2}(x+2 \,y)$
• B. $\frac{k}{2}(2\, x+y)$
• C. $k y(x+2\, y)$
• D. $\frac{k y}{2}(2 \,x+y)$

Answer 1

Answer: (D)

According to the question
In the first condition,
$ W_{1}=\frac{1}{2} k x^{2}\,\,\,...(i)$
In the second condition
$W_{2}=\frac{1}{2} k(x+y)^{2}\,\,\,...(ii)$
Work for $ x$ to $y$ expansion
$=\frac{1}{2} k(x+y)^{2}-\frac{1}{2} k x^{2}$
$=\frac{1}{2} k\left[x^{2}+y^{2}+2 \,x y-x^{2}\right]$
$=\frac{1}{2} k\left[y^{2}+2\, x y\right]$
$=\frac{1}{2} k y[y+2\, x]$