Question

Fig shows a tappered wire of length L under an external force F. The radius at end O and A is 2r and r respectively. Y is Young’s modulus of material of wire. A normal external force F is applied at end A. The increase in length of wire is

(1) \(\frac{FL}{\pi Yr^2}\)

(2) \(\frac{FL}{2\pi Yr^2}\)

(3) \(\frac{2FL}{3\pi Y r^2}\)

(4) \(\frac{3FL}{2 \pi Yr^2}\)

Answer 1

 (2) \(\frac{FL}{2\pi Yr^2}\)

The rate of change of radius along the length of wire

Consider a small element of wire define by y and y + dy as shown in Fig

r₁ = The radius of the element considered = [2r + ay]

Let dx by be the change in length of element of original length dy due to applied force F. Then