Q. An elastic string 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) $\frac{1}{2}ky^{2}$
B) $\frac{1}{2}k(x^{2} + y^{2})$
C) $\frac{1}{2}k(x + y)^{2}$
D) $\frac{1}{2}ky(2x + y)$
Explanation
(d) Elastic force in string is conservative in nature
$W = - \mathrm{\Delta}V_{1}$ where $W =$ work done by elastic force of string
$W = - \left( V_{f} - V_{i} \right) = V_{i} - V_{f}$ or $W = \frac{1}{2}kx^{2} - \frac{1}{2}k(x + y)^{2}$
or $W = \frac{1}{2}kx^{2} - \frac{1}{2}k(x^{2} + y^{2} + 2xy)$
$= \frac{1}{2}kx^{2} - \frac{1}{2}kx^{2} - \frac{1}{2}ky^{2} - \frac{1}{2}k(2xy) = - kxy - \frac{1}{2}ky^{2}$
$= \frac{1}{2}ky( - 2x - y)$
The work done against elastic force is
$W_{ext} = - W = \frac{ky}{2}(2x + y)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.