Chapter 5
BUOYANCY AND
STABILITY
5.1 Objectives
1. Write the equation for the buoyant force.
2. Analyze the case of bodies floating on a fluid.
3. Use the principle of static equilibrium to solve for the forces
4. Define the conditions that must be met for a body to be
stable when completely submerged in a fluid.
5. Define the stability conditions for a body when floating on a fluid.
6. Define the term metacenter and compute its location.
5.2 BUOYANCY
A body in a fluid, whether floating or submerged, is buoyed up by
a force equal to the weight of the fluid displaced.
The buoyant force acts vertically upward through the centroid of
the displaced volume.
➭ Buoyant Force
Fb = Buoyant force
gf = Specific weight of the fluid
Vd = Displaced volume of the fluid
Example Problem
5.1
A cube 0.50 m on a side is made of bronze having a specific weight of
86.9 kN/m3. Determine the magnitude and direction of the force
required to hold the cube in equilibrium when completely submerged (a)
in water and (b) in mercury. The specific gravity of mercury is 13.54
Example Problem
5.1
A cube 0.50 m on a side is made of bronze having a specific weight of
86.9 kN/m3. Determine the magnitude and direction of the force
required to hold the cube in equilibrium when completely submerged (a)
in water and (b) in mercury. The specific gravity of mercury is 13.54
Example Problem
5.2
A certain solid metal object weighs 60 lb when measured in the
normal manner in air, but it has such an irregular shape that it is
difficult to calculate its volume by geometry. Use the principle of
buoyancy to calculate its volume and specific weight.
Example Problem
5.3
A cube 80 mm on a side is made of a rigid foam material and
floats in water with 60 mm of the cube below the surface.
Calculate the magnitude and direction of the force required to
hold it completely submerged in glycerin, which has a specific
gravity of 1.26
Example Problem
5.3
A cube 80 mm on a side is made of a rigid foam material and
floats in water with 60 mm of the cube below the surface.
Calculate the magnitude and direction of the force required to
hold it completely submerged in glycerin, which has a specific
gravity of 1.26
Example Problem
5.4
A brass cube 6 in on a side weighs 67 lb. We want to hold this
cube in equilibrium under water by attaching a light foam buoy to
it. If the foam weighs 4.5 lb/ft3, what is the minimum required
volume of the buoy?
5.3 BUOYANCY MATERIALS
The buoyancy material should typically have the following properties:
■ Low specific weight and density
■ Little or no tendency to absorb the fluid
■ Compatibility with the fluid in which it will operate
■ Ability to be formed to appropriate shapes
■ Ability to withstand fluid pressures to which it will be subjected
■ Abrasion resistance and damage tolerance
■ Attractive appearance
5.4 STABILITY OF COMPLETELY
SUBMERGED BODIES
➭ Condition of Stability for Submerged Bodies
The condition for stability of bodies completely submerged in a
fluid is that the center of gravity of the body must be below the
center of buoyancy
ycb>ycg the body stable
5.5 STABILITY OF FLOATING
BODIES
➭ Condition of Stability for Floating Bodies
A floating body is stable if its center of gravity is below the
metacenter
I is the least moment of
inertia of a horizontal
section of the body taken
at the surface of the fluid
5.5 STABILITY OF FLOATING
BODIES
Procedure for Evaluating the Stability of Floating Bodies
1. Determine the position of the floating body
2. Locate the center of buoyancy, cb; compute the ycb.
3. Locate the center of gravity, cg; compute ycg
4. Compute the smallest moment of inertia I for that shape.
5. Compute the displaced volume Vd.
6. Compute MB = I/Vd.
7. Compute ymc = ycb + MB.
8. If ymc > ycg, the body is stable.
9. If ymc < ycg, the body is unstable
Example Problem
5.5
⚫ Figure 5.11 shows a flatboat hull that, when fully loaded, weighs
150 kN. Parts (b)–(d) show the top, front, and side views of the
boat, respectively. Note the location of the center of gravity, cg.
Determine whether the boat is stable in fresh water
Example Problem
5.5
Example Problem
5.5
Example Problem
5.6
⚫ A solid cylinder is 3.0 ft in diameter, 6.0 ft
high, and weighs 1550 lb. If the cylinder is
placed in oil (sg = 0.90) with its axis vertical,
would it be stable?
Example Problem
5.6
⚫ A solid cylinder is 3.0 ft in diameter, 6.0 ft
high, and weighs 1550 lb. If the cylinder is
placed in oil (sg = 0.90) with its axis vertical,
would it be stable?
5.6 DEGREE OF STABILITY
⚫ One measure of relative stability is called the metacentric
height, defined as the distance to the metacenter above the
center of gravity and called MG
⚫ An object with a larger metacentric height is more stable than
one with a smaller value
⚫ vessels should have a minimum value of MG of 1.5 ft (0.46 m).
Large ships should have MG 7 3.5 ft (1.07 m).
