If work W is required to stretch a spring a distance x from its unstretched length, what work (in terms of W) is required to stretch the spring an ?additional? distance x?

Solution 24DQ Step 1: Consider a spring having spring constant, “k” is stretched to a distance “x” from its initial length “l”. Then, the restoring force is, F = - k x The work done would be, W = F x We should consider the magnitude of the force while calculating the work done because, the restoring force and force applied is having equal magnitude. 2 Therefore, W = kx × x = kx Step 2: When in order to stretch the spring an additional distance of “x”, we should apply a force, F ’ = k × (x+x) = 2kx The work done would be, W ’ = F ’ × 2x = 2kx × 2x = 4kx = 4 W, (we know that, Fx = W) Therefore, in order to stretch the spring for a distance of 2x, the work done would be four times that of the work needed to stretch the spring for “x” distance.