CHAPTER 12
SHIP STABILITY AND BUOYANCY
TRIGONOMETRY
Learning Objectives: Recall the terminology used for
ship stability; the laws of physics and trigonometry used
Trigonometry is the study of triangles and the
to determine stability and buoyancy of a ship; and the
interrelationship of the sides and the angles of a
effects of buoyancy, gravity, and weight shifts on ship
triangle. In determining ship stability, only that part of
trigonometry pertaining to right triangles is used.
stability.
There is a fixed relationship between the angles of a
Under the guidance of the damage control
right triangle and the ratios of the lengths of the sides of
assistant, damage control personnel provide the first
the triangle. These ratios are known as trigonometric
line of defense to ensure your ship is as seaworthy as
functions and have been given the following names:
possible. Your responsibilities may include preparing
daily draft reports, taking soundings, or perhaps you
sine, cosine, tangent, cotangent, secant, and cosecant.
may stand watch operating a ballasting console.
The three trigonometric functions required for ship
stability work are the sine, cosine, and tangent. Figure
In this chapter, you will be introduced to the laws
12-1 shows these trigonometric relations.
of mathematics and physics used to determine the
buoyancy and stability of a ship. Also, there are various
Sine
engineering and mathematical principles that you will
become familiar with as you study this chapter.
In trigonometry, angles are represented by the
Detailed information on these subjects is provided in
Greek letter theta (θ). The sine of an angle θ,
the Naval Ships' Technical Manual (NSTM), chapter
abbreviated as sin θ, is the ratio expressed when the
079, volume 1, and in NSTM, chapter 096. You can find
side of a right triangle opposite the angle θ is divided
additional information on these subjects in
publications you will find listed in the Damage
b y t h e h y p o t e n u s e . F i g u r e 1 2 - 1 s h ow s t h e s e
Controlman Advancement Handbook.
trigonometric relations.
Therefore, referring to figure 12-1:
PRINCIPLES OF STABILITY
sin θ = y/r, or the altitude (y) divided by the
Learning Objectives: Recall the basic functions of
hypotenuse (r)
trigonometry, the terminology used for ship stability,
If the hypotenuse (r) is also the radius of a circle,
the effects of buoyancy and gravity on ship stability, and
point P moves along the circumference as the angle
the effects of weight shifts on ship stability.
changes in size. As angle θ increases, side y increases
To comprehend the principles of ship stability
in length while the length of the hypotenuse (or radius)
fully, you must have a basic understanding of
remains the same. Therefore, the value of the sine
trigonometry and the functions of right triangles.
increases as the angle increases. Changes in the value
Generally speaking, the weight of a ship in the water is
of the sine corresponding to changes in the size of the
"pushing" straight down, and the seawater that it
angle are shown on the sine curve shown in figure 12-2.
displaces is "pushing" straight back up. When no other
forces are acting on the ship, all these forces cancel
On the sine curve, the size of the angle is plotted
each other out and equilibrium exists. However, when
horizontally and the value of the sine vertically.
the center of gravity moves from directly above the
At any angle, the vertical height between the
center of buoyancy, there is an "inclining moment."
baseline and the curve is the value of the sine of the
When this occurs, this force is considered to be at right
angle. This curve shows that the value of the sine at 30
angles to the forces of gravity and buoyancy. An
is half of the value of the sine at 90. At 0, sin θ equals
understanding of trigonometry is required to
zero. At 90, sin θ equals one.
understand the effects and results of these actions.
12-1
