A uniform wire of mass $m$ and length $L$ hangs vertically from a rigid support, and a block of mass $2m$ is attached to its lower end. The ratio of the extension of the lower half of the wire to that of the upper half is
Answer: (D) $9 : 11$
Concept. The tension varies along the wire because each part also supports the wire below it. The extension of a small element $dx$ is $\dfrac{T\,dx}{AY}$, so the extension of a section is $\dfrac{1}{AY}\int T\,dx$.
Step 1. Tension. At height $x$ above the lower end:
$$T(x) = 2mg + \frac{mg}{L}x$$
Step 2. Lower half ($0$ to $\frac L2$).
$$\int_0^{L/2}T\,dx = 2mg\cdot\frac L2 + \frac{mg}{L}\cdot\frac{L^2}{8} = \frac98mgL$$
Step 3. Upper half ($\frac L2$ to $L$).
$$\int_{L/2}^{L}T\,dx = 2mg\cdot\frac L2 + \frac{mg}{2L}\left(L^2 - \frac{L^2}{4}\right) = \frac{11}{8}mgL$$
Step 4. Ratio. $\dfrac98 : \dfrac{11}{8} = 9 : 11$.
Answer. $9 : 11$ (Option D). Without the block the ratio would be $1 : 3$; with a block of mass $m$ it would be $5 : 7$.
Solution by Sreeraj P, M.Sc Physics