Question

A spring of unstretched length $l$ and force constant $k$ is stretched by a small length $x$. It is further stretched by another small length $y$. The work done in the second stretching is

• A. $\frac{1}{2}k{x}^2$
• B. $\frac{1}{2}k(x^2+y^2)$
• C. $\frac{1}{2}ky(2x+y)$
• D. $\frac{1}{2}kx(x+2y)$

Answer 1

The correct option is C $\frac{1}{2}ky(2x+y)$

When the spring is stretched by the spring force which is conservative in nature, work done by spring force is
$W=-\Delta U=-(U_f-U_i)=U_i-U_f$
$=\frac{1}{2}k{x}^2-\left(\frac{1}{2}k\right(x+y{)}^2)$
$=\frac{1}{2}k{x}^2-\frac{1}{2}k{x}^2-\frac{1}{2}k{y}^2-kxy$
$=\frac{-1}{2}k{y}^2-kxy=\frac{-1}{2}ky(y+2x)$
Work done by external agent is negative of this.
Hence, the correct answer is option (c).