Exercises

1. A hollow vessel floats in a basin. If, as a consequence of a leak, water flows slowly into the vessel, how will the level of the water in the basin be affected? (Lamb 1928.)

2. A hollow spherical shell made up of material of specific gravity $s>1$ has external and internal radii $a$ and $b$ , respectively. Demonstrate that the sphere will only float in water if
   $$\frac{b}{a}> \left(1-\frac{1}{s}\right)^{1/3}.$$

3. Show that the equilibrium of a solid of uniform density floating with an edge or corner just emerging from the water is unstable. (Lamb 1928.)

4. Prove that if a solid of uniform density floats with a flat face just above the waterline then the equilibrium is stable. (Lamb 1928.)

5. Demonstrate that a uniform solid cylinder floating with its axis horizontal is in a stable equilibrium provided that its length exceeds the breadth of the waterline section. [Hint: The cylinder is obviously neutrally stable to rotations about its axis, which means that the corresponding metacentric height is zero.] (Lamb 1928.)

6. Show that a uniform solid cylinder of radius $a$ and height $h$ can float in stable equilibrium, with its axis vertical, if $h/a<\sqrt{2}$ . If the ratio $h/a$ exceeds this value, prove that the equilibrium is only stable when the specific gravity of the cylinder lies outside the range
   $$\frac{1}{2}\left(1\pm \sqrt{1-2\,\frac{a^{\,2}}{h^{\,2}}}\,\right).$$

7. A uniform, thin, hollow cylinder of radius $a$ and height $h$ is open at both ends. Assuming that $h>2\,a$ , prove that the cylinder cannot float upright if its specific gravity lies in the range
   $$\frac{1}{2}\pm \sqrt{\frac{1}{4}-\frac{a^{\,2}}{h^{\,2}}}.$$
   (Lamb 1928.)

8. Show that the cylinder of the preceding exercise can float with its axis horizontal provided
   $$\frac{h}{2\,a}> \sqrt{3}\,\sin(s\,\pi),$$
   where $s$ is the specific gravity of the cylinder. (Lamb 1928.)

9. Prove that any segment of a uniform sphere, made up of a substance lighter than water, can float in stable equilibrium with its plane surface horizontal and immersed. (Lamb 1928.)

10. A vessel carries a tank of oil, of specific gravity $s$ , running along its length. Assuming that the surface of the oil is at sea level, show that the effect of the oil's fluidity on the rolling of the vessel is equivalent to a reduction in the metacentric height by $A\,\kappa^{\,2}\,s/V$ , where $V$ is the displacement of the ship, $A$ the surface-area of the tank, and $\kappa$ the radius of gyration of this area. In what ratio is the effect diminished when a longitudinal partition bisects the tank? (Lamb 1928.)

11. Find the stable equilibrium configurations of a cylinder of elliptic cross-section, with major and minor radii $a$ and $b<a$ , respectively, made up of material of specific gravity $s$ , which floats with its axis horizontal.

12. A cylindrical tank has a circular cross-section of radius $a$ . Let the center of gravity of the tank be located a distance $c$ above its base. Suppose that the tank is pivoted about a horizontal axis passing through its center of gravity, and is then filled with fluid up to a depth $h$ above its base. Demonstrate that the position in which the tank's axis is upright is unstable for all filling depths provided
    $$c^{\,2}< \frac{1}{2}\,a^{\,2}.$$
    Show that if $c^{\,2}>(1/2)\,a^{\,2}$ then the upright position is stable when $h$ lies in the range
    $$c\pm \sqrt{c^{\,2}-a^{\,2}/2}.$$

13. A thin cylindrical vessel of cross-sectional area $A$ floats upright, being immersed to a depth $h$ , and contains water to a depth $k$ . Show that the work required to pump out the water is $\rho_0\,A\,k\,(h-k)\,g$ . (Lamb 1928.)

14. A sphere of radius $a$ is just immersed in water that is contained in a cylindrical vessel of radius $R$ whose axis is vertical. Prove that if the sphere is raised just clear of the water then the water's loss of potential energy is
    $$W\,a\left(1-\frac{2}{3}\,\frac{a^{\,2}}{R^{\,2}}\right),$$
    where $W$ is the weight of the water originally displaced by the sphere. (Lamb 1928.)

15. A sphere of radius $a$ , weight $W$ , and specific gravity $s>1$ , rests on the bottom of a cylindrical vessel of radius $R$ whose axis is vertical, and which contains water to a depth $h>2\,a$ . Show that the work required to lift the sphere out of the vessel is less than if the water had been absent by an amount
    $$\left(h-a-\frac{2}{3}\,\frac{a^{\,3}}{R^{\,2}}\right)\frac{W}{s}.$$
    (Lamb 1928.)

16. A lead weight is immersed in water that is steadily rotating at an angular velocity $\omega$ about a vertical axis, the weight being suspended from a fixed point on this axis by a string of length $l$ . Prove that the position in which the weight hangs vertically downward is stable or unstable depending on whether $l< g/\omega^{\,2}$ or $l>g/\omega^{\,2}$ , respectively. Also, show that if the vertical position is unstable then there exists a stable inclined position in which the string is normal to the surface of equal pressure passing though the weight.

17. A thin cylindrical vessel of radius $a$ and height $H$ is orientated such that its axis is vertical. Suppose that the vessel is filled with liquid of density $\rho$ to some height $h<H$ above the base, spun about its axis at a steady angular velocity $\omega$ , and the liquid allowed to attain a steady state. Demonstrate that, provided $\omega^{\,2}\,a^{\,2}/g < 4\,h$ and $\omega^{\,2}\,a^{\,2}/g< 4\,(H-h)$ , the net radial thrust on the vertical walls of the vessel is
    $$\pi\,a\,h^{\,2}\,\rho\,g\left(1+ \frac{\omega^{\,2}\,a^{\,2}}{4\,g\,h}\right)^2.$$

18. A thin cylindrical vessel of radius $a$ with a plane horizontal lid is just filled with liquid of density $\rho$ , and the whole rotated about a vertical axis at a fixed angular velocity $\omega$ . Prove that the net upward thrust of the fluid on the lid is
    $$\frac{1}{4}\,\pi\,a^{\,4}\,\rho\,\omega^{\,2}.$$

19. A liquid-filled thin spherical vessel of radius $a$ spins about a vertical diameter at the fixed angular velocity $\omega$ . Assuming that the liquid co-rotates with the vessel, and that $\omega^{\,2}>g/a$ , show that the pressure on the wall of the vessel is greatest a depth $g/\omega^{\,2}$ below the center. Also prove that the net normal thrusts on the lower and upper hemispheres are
    $$\frac{5}{4}\,M\,g+ \frac{3}{16}\,M\,\omega^{\,2}\,a,$$
    and
    $$\frac{1}{4}\,M\,g - \frac{3}{16}\,M\,\omega^{\,2}\,a,$$
    respectively, where $M$ is the mass of the liquid.

20. A closed cubic vessel filled with water is rotating about a vertical axis passing through the centers of two opposite sides. Demonstrate that, as a consequence of the rotation, the net thrust on a side is increased by
    $$\frac{1}{6}\,a^4\,\rho\,\omega^{\,2},$$
    where $a$ is the length of an edge of the cube, and $\omega$ the angular velocity of rotation.

21. A closed vessel filled with water is rotating at constant angular velocity $\omega$ about a horizontal axis. Show that, in the state of relative equilibrium, the constant pressure surfaces in the water are circular cylinders whose common axis is a height $g/\omega^{\,2}$ above the axis of rotation. (Batchelor 2000.)

22. Verify Equations (2.95)-(2.97) and (2.99)-(2.101).

23. Consider a homogeneous, rotating, liquid body of mass $M$ , mean radius $a_0$ , and angular velocity $\omega$ , whose outer boundary is a Maclaurin spheroid of eccentricity $e$ .
    1. Demonstrate that
       $$e\simeq \sqrt{\frac{5}{2}}\,\frac{\omega}{(G\,M/a_0^{\,3})^{1/2}}$$
       in the low rotation limit, $\omega\ll (G\,M/a_0^{\,3})^{1/2}$ . Hence, show that $e=0.09262$ for the case of a homogeneous body with the same mass and volume as the Earth, which rotates once every 24 hours.

    2. Show that the critical angular velocity at which the bifurcation to the sequence of Jacobi ellipsoids takes place is
       $$\omega = 0.5298\,(G\,M/a_0^{\,3})^{1/2},$$
       and occurs when $e=0.81267$ . Hence, show that, for the case of a homogeneous body with the same mass and volume as the Earth, the bifurcation would take place at a critical rotation period of $2\,{\rm h}\,39\,{\rm m}$ .

    3. Demonstrate that the maximum angular velocity consistent with a spheroidal shape is
       $$\omega = 0.5805\,(G\,M/a_0^{\,3})^{1/2},$$
       and occurs when $e=0.92995$ . Hence, show that, for the case of a homogeneous body with the same mass and volume as the Earth, this maximum velocity corresponds to a minimum rotation period of $2\,{\rm h}\,25\,{\rm m}$ .