Question

A cylinder of mass m height h, radius r, when put in water, it floats with half of its volume submerged. The cylinder is placed in a cylindrical vessel of inner radius R = 2r, on a spring of force constant

$k = \frac{mg}{(\frac{\sqrt{3}}{2}h)}$, such that cylinder is inside water by $\frac{h}{3}$ as shown in the figure. The minimum energy required to push the cylinder such that it just completely immersed in water is

• A. $\frac{mgh}{6}$
• B. $mgh(\frac{\sqrt{3}+4}{12})$
• C. $\frac{mgh}{2}(1+\sqrt{3})$
• D. $\frac{mgh}{6\sqrt{3}}(1+\sqrt{3})$

Answer 1

Answer: (B)

$mgh(\frac{\sqrt{3}+4}{12})$

Answer 2

1. Write the extra energy (increase in spring energy + increase in gravitational energy of water – drop in the cylinder’s gravitational energy). Because the cylinder “displaces” extra water when pushed a distance u, the water level in a vessel of area 4A₍c₎ rises by y = u/5.

2. Hence, the net extra energy is $\Delta U(u) = \frac{1}{2} k u^2 + \frac{1}{2}(\rho g \cdot 4A_c)(\frac{u}{5})^2 - mg u$, with $k = \frac{2mg}{\sqrt{3}h}$ [since $mg = \rho g A_c (\frac{h}{2})$].

3. Noting that “just complete immersion” requires $u = \frac{5h}{6}$, one finds that the maximum energy “barrier” (obtained by a standard procedure) is $W_{min} = mgh(\frac{\sqrt{3}+4}{12})$.