- 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.)
- A hollow spherical shell made up of material of specific gravity
has external and internal
radii
and
, respectively. Demonstrate that the sphere will only float in water if
- 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.)
- Prove that if a solid of uniform density floats with a flat face just above the waterline then the
equilibrium is stable. (Lamb 1928.)
- 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.)
- Show that a uniform solid cylinder of radius
and height
can float in stable equilibrium, with its
axis vertical, if
. If the ratio
exceeds this value, prove that the equilibrium is
only stable when the specific gravity of the cylinder lies outside the range
- A uniform, thin, hollow cylinder of radius
and height
is open at both ends. Assuming that
, prove that the
cylinder cannot
float upright if its specific gravity lies in the range
- Show that the cylinder of the preceding exercise can float with its axis horizontal provided
- 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.)
- A vessel carries a tank of oil, of specific gravity
, 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
,
where
is the displacement of the ship,
the surface-area of the tank, and
the radius
of gyration of this area. In what ratio is the effect diminished when a longitudinal partition bisects the
tank? (Lamb 1928.)
- Find the stable equilibrium configurations of a cylinder of elliptic cross-section, with major and minor radii
and
, respectively, made up of material of specific gravity
, which floats with its axis horizontal.
- A cylindrical tank has a circular cross-section of radius
.
Let the center of gravity of the tank be located a
distance
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
above its base. Demonstrate that the position in which the
tank's axis is upright is unstable for all filling depths provided
- A thin cylindrical vessel of cross-sectional area
floats upright, being immersed to a depth
, and
contains water to a depth
. Show that the work required to pump out the water is
.
(Lamb 1928.)
- A sphere of radius
is just immersed in water that is contained in a cylindrical vessel of radius
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
- A sphere of radius
, weight
, and specific gravity
, rests on the bottom of
a cylindrical vessel of radius
whose axis is vertical, and which contains
water to a depth
. 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
- A lead weight is immersed in water that is steadily rotating at an angular velocity
about a vertical axis, the
weight being suspended from a fixed point on this axis by a string of length
. Prove that the
position in which the weight hangs vertically downward is stable or unstable depending on whether
or
,
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.
- A thin cylindrical vessel of radius
and height
is orientated such that its axis is vertical. Suppose that the
vessel is filled with liquid of density
to
some height
above the base, spun about its axis at a steady angular velocity
, and the liquid
allowed to attain a steady state. Demonstrate that, provided
and
,
the net radial thrust on the vertical walls of the vessel is
- A thin cylindrical vessel of radius
with a plane horizontal lid is just filled with liquid of density
, and
the whole rotated about a vertical axis at a fixed angular velocity
. Prove that the net upward thrust of the
fluid on the lid is
- A liquid-filled thin spherical vessel of radius
spins about a vertical diameter at the fixed angular velocity
.
Assuming that the liquid co-rotates with the vessel, and that
, show that the pressure on the wall of the
vessel is greatest a depth
below the center. Also prove that the net normal thrusts on the lower
and upper hemispheres are
- 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
- A closed vessel filled with water is rotating at constant angular velocity
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
above the axis of rotation. (Batchelor 2000.)
- Verify Equations (2.95)-(2.97) and (2.99)-(2.101).
- Consider a homogeneous, rotating, liquid body of mass
, mean radius
, and angular velocity
, whose
outer boundary is a Maclaurin spheroid of eccentricity
.
- Demonstrate that
- Show that the critical angular velocity at which the bifurcation to the sequence of Jacobi ellipsoids
takes place is
- Demonstrate that the maximum angular velocity consistent with a spheroidal shape is

- Demonstrate that