**(A)** \(\frac{mg}{3Ka}\)

Bulk modulus is given as,

K = \((\frac{-dP}{dV/V})\)

where negative sign indicates volume decreases with increase in pressure.

∴ Fractional decrease in volume will be,

\(\frac{dV}{V}=\frac{dP}{K}\) If area of cross-section of cylinder is a, then,

\(\frac{dV}{V}=\frac{dP}{K}\) If area of cross-section of cylinder is a, then,

**Also,**

Equating equations (i) and (ii),

\(\frac{dr}{r}=\frac{mg}{3Ka}\)