Stability and the Center of Gravity

Center of Gravity

When studying objects, forces and their inter-relationships we often tend to represent a body by a point for theoretical purposes of calculations relating to that body. This certainly is a convenient method of representation but we know that in actual situation there is hardly any body which is actually the size of a point, so how do we go about it?

Well we simply imagine that if the whole mass of the body were to be concentrated at a single point where would that point be? That imaginary point is known as the center of mass of the body. We also know that whether any other force acts on a body or not, the force of gravity certainly acts on it (remember we are not dealing with space stuff here but objects on the earth hence gravity is always present). Even this force of gravity can be assumed to act at the center of mass and hence it is also known as the center of gravity of the body which means a point where the whole weight of the body seems to be concentrated.

How to find out the center of gravity?

A body can be assumed to be made of small pieces joined together and let there be small pieces of mass “m” each with the distance “d” from the center of gravity. The center of gravity can be calculated as the summation of the moments of all the masses divided by the total mass of the body i.e.

R = (∑ m * d)/∑m

But you need not do this for all bodies for the well defined geometrical shapes the center of gravity coincides with the geometrical center (it is also known as centroid) of the body and here are a few cases

* In case of circle, the center of gravity is at its center

* In case of square and rectangle, the center of gravity is at the intersection of two diagonals

Change in Mass and change in CG

Since center of gravity depends on mass, certainly any change in mass would result in a change of center of gravity of the body. Now if you are thinking in terms of conservation of mass I do not mean that mass is created or destroyed but I simply mean that some weight is either place on or removed from the object under discussion. The mathematical analysis of such a case suggests that whenever some amount of mass is either added (or removed) from an object the center of gravity of the body move towards (or away from) the center of gravity of the body by a distance which is directly proportional to the mass added (or removed) and its distance from the original center of gravity of the body and inversely proportional to the total increased (or decreased) mass of the combined bodies.

Mathematically this can be expressed as follows

CG = m * s/M + m

Where “m” is the mass added (or removed), M is original mass of the body and “s” is the distance between the two centers of gravities of the separate bodies.

Corollary: another thing which can be concluded from the above equation is that if the mass is neither removed nor added but only moved from one position to other on the parent body, the new center of gravity is given by the same equation except that M remains unchanged, therefore

CG = m * s/M

How is this relevant to a ship?

If you just remembered that we are talking about ship stability here then let me explain how this concept is useful in ship calculations. We know that ships contain huge cargo hatches from where cargo is removed or added in bulk quantities during loading or discharging operations. This change of mass will cause a change in the center of gravity of the entire ship and hence you can start to see a connection between the stuff studied till now and its application to stability. We will proceed with this study in our next article.