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. \(\dfrac{1}{2}Ky^2\)
• B. \(\dfrac{1}{2}K y(2x+y)\)
• C. \(\dfrac{1}{2}K(x^2+y^2)\)
• D. \(\dfrac{1}{2}K(x+y)^2\)

Hint:

Elastic string work: \[ W = \frac{1}{2}k(x_2^2 - x_1^2) \] Always integrate tension over extension.

Answer 1

The Correct Option is B

Step 1: Tension in elastic string: \[ T = k(\text{extension}) \]
Step 2: Work done in stretching from \(x\) to \(x+y\): \[ W = \int_x^{x+y} kx\,dx \]
Step 3: Integrating: \[ W = \frac{1}{2}k[(x+y)^2 - x^2] \]
Step 4: Simplify: \[ W = \frac{1}{2}k y(2x+y) \]