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

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

trigonometric relations.

Therefore, referring to figure 12-1:

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