Question

A block of mass m is attached rigidly with a light spring of force constant K. The other end of the spring is fixed to a wall. If block is displaced by a distance x then find the work done on the block by the spring for this range. K is spring constant and x is displacement of the block.

• A. $-\frac{1}{3}K{x}^2$
• B. $-\frac{1}{2}K{x}^2$
• C. $-\frac{1}{6}K{x}^2$
• D. $Zero$

Answer 1

The correct option is B $-\frac{1}{2}K{x}^2$


Since F = kx, therefore force varies with displacement. This force has tendency to bring the block to its equilibrium point hence it is opposite to the displacement.
At the moment when elongation in the spring is x, spring force F = kx.
For infinitesimal displacement (dx) this force is supposed to be constant. Therefore work done by this force for the displacement dx is given by
$dw=\stackrel{\rightharpoonup }{F}.\stackrel{\rightharpoonup }{dx}=-kxdxW=\int dw=-{\int }_0^{x}kxdxW=\frac{1}{2}K{x}^2$