Question

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. $1/2 Ky^2$
• B. $1/2 Ky(2x + y)$
• C. $1/2 K(x^2 + y^2)$
• D. $1/2 K(x +y)^2$

Answer 1

Answer: (B)

In the string elastic force is conservative in nature.
$\therefore W =-\Delta U$
Work done by elastic force of string.
$W =-\left( U _{ F }- U _{ i }\right)= U _{ i }- U _{ F }$
$W =\frac{1}{2} kx ^{2}-\frac{ k }{2}( x + y )^{2}$
$=\frac{1}{2} kx ^{2}-\frac{1}{2} k \left( x ^{2}+ y ^{2}+2 xy \right)$
$=\frac{1}{2} kx ^{2}-\frac{1}{2} ky ^{2}-\frac{1}{2} kx ^{2}-\frac{1}{2} k (2 xy )$
$=- kxy -\frac{1}{2} ky ^{2}$
Therefore, the work done against elastic force
$W _{\text {external }}=- W =\frac{ ky }{2}(2 x + y )$.