
STATICS  CASE STUDY



Introduction

Geometry Diagram 

In order to determine a submarine's buoyancy characteristics, designers
must accurately determine the centroid, center of mass, and center of
gravity for various parts of the sub.
What is known:
 The radius of the nose cone is given by
r(x) = B  Cx^{D}.
 The length of the nose cone is 100 ft.
 The base of the nose cone has a radius 20 ft.
 The shape exponent D is 4, but as a first approximation, the designers
use D = 1 in order to simplify the mathematics.
 The density of the nose cone varies linearly from 1 slug/ft^{3}
at the base to 2 slug/ft^{3} at the tip.






Question

Submarine Path Graphic


As a first approximation, assume the nose code is a simple cone shape (D = 1 in the above equation). Where is the centroid, center of mass, and center of gravity?




Approach


 Use symmetry to determine the y and z locations of the centroid, center of mass, and center of gravity.
 Solve for the x location by integrating the appropriate quantities from the base to the tip of the nose cone.




