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/ft³ at the base to 2 slug/ft³ at the tip.

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?

• 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.