# want some to help me in that question

Discussion in 'Stability' started by abdelrhman shah, Dec 15, 2014.

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### abdelrhman shahJunior Member

I want know. Why is the Initial metacentric height (GM).is the point of intersection when a tangent drawn at the origin and the purpendicular at 57.3°

why especially 57.3

and why the tangent drawn ....

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### CanracerSenior Member

When you say tangent, do you really mean a straight line? When I hear the word tangent, I think of a 90 deg angle.

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### TANSLSenior Member

Refers to a straight line that is tangent to the curve of GZ at the origin.
Canracer, I think you have to learn as much or more than he.

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The reason is because the 1st assumption is that the GZ curve is drawn for small angles of inclination. Thus the GZ = GM.Sin(theta). Where theta is small the angle is measured in radian.s Therefore 1 rad = 57.3 degrees.

And since this is true at small angles only, the line is drawn tangentially from/to the origin at 57.3 degree. Because at larger angles of inclination this assumption is no longer valid.

So that line passes through the origin, and the GZ curve for a few degrees is considered a straight line (since in rads). How straight of course depends upon the shape of the hull under investigation.

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### abdelrhman shahJunior Member

Thanks alot.....but I still feeling confused. ..... or iam so stupid to under stand it

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### CanracerSenior Member

I don't see that line to be drawn as a tangent to the curve. A tangent would be 90 degrees to the radius, right?

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### TANSLSenior Member

"90 degrees to the radius" drawn from the center of curvature of the curve. Do you know where the center of curvature of the curve is?. I do not.
The process is reverse, the straight line is drawn from the origin to the point of coordinates (1 radian, initial GMt) and GZ value curve must be tangent to this line.

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### cmckessonNaval Architect

Abdel

Try this:

What is the slope of the GZ curve? It is d GZ/ d (theta)

Calculate the value of that derivative when GZ = GM sin(theta)

At theta=0, what is the value? In other words, at theta=0, what is the slope of the GZ curve?

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### abdelrhman shahJunior Member

I know that 57.3 = 1 radian....why we draw line from this point ....why especially 57.3

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### Eric SponbergSenior Member

Abdelrhman Shah, Ad Hoc gave the explanation above. At small angles of heel, the righting arm, GZ = GMSin(theta). And also at small angles, Sin(theta) = theta when theta is expressed in radians. Therefore, GZ = GM x theta at small angles.

So on your diagram in your first post, at small angles of heel, you can calculate the angle between the blue line or the red line, which we'll call alpha, with the equation Tan(alpha) = GZ/theta. You show the horizontal axis in degrees of heel, but in the equation and at small angles, theta is actually in radians. In the paragraph above, we see that GZ = GM x theta, and if we divide both sides of the equation by theta, we get GZ/theta = GM. Substituting GZ/theta for GM we get:

Tan(alpha) = GM

Another way to look at this is to realize that if theta equals zero, then the tangent equation becomes undefined. That is, Tan(alpha) = GZ/0 = infinity. But if theta = 1 (that is, if it equals one radian = 57.3 degrees) then we have Tan(alpha) = GZ/1 = GZ which is still equal to GM. That is why when you plot the tangent to the blue curve by drawing in the red line, and you are using degrees as the labels on the horizontal axis, you have to go across one radian = 57.3 degrees and the vertical erected at that point will cross the red line at the value of GM, which is really GZ/1.

Does that make it any clearer?

Eric

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### abdelrhman shahJunior Member

Yeah I finally understood.....thanks very much

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