Editing Center of mass (section)

==Determination{{anchor|Locating}}==
{{see also|Centroid#Determination}}
[[File:Center gravity 2.svg|thumb|Plumb line method]]
The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center of gravity in the parallel gravity field near the earth's surface.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis.  Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder.  In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere.  In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.<ref>[https://feynmanlectures.caltech.edu/I_19.html#Ch19-S1-p3 The Feynman Lectures on Physics Vol. I Ch. 19: Center of Mass; Moment of Inertia]</ref>

===In two dimensions===
An experimental method for locating the center of mass is to suspend the object from two locations and to drop [[plumb line]]s from the suspension points. The intersection of the two lines is the center of mass.{{sfn|Kleppner|Kolenkow|1973|pp=119–120}}

The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers.{{sfn|Feynman|Leighton|Sands|1963|pp=19.1–19.2}} This method can even work for objects with holes, which can be accounted for as negative masses.{{sfn|Hamill|2009|pp=20–21}}

A direct development of the [[planimeter]] known as an integraph, or integerometer, can be used to establish the position of the [[centroid]] or center of mass of an irregular two-dimensional shape. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to compare with the required [[Displacement (ship)|displacement]] and [[center of buoyancy]] of a ship, and ensure it would not capsize.<ref>{{cite web|title=The theory and design of British shipbuilding |page=3 |url=http://www.ebooksread.com/authors-eng/amos-lowrey-ayre/the-theory-and-design-of-british-shipbuilding-hci/page-3-the-theory-and-design-of-british-shipbuilding-hci.shtml|work=Amos Lowrey Ayre|access-date=20 August 2012}}</ref>{{sfn|Sangwin|2006|p=7}}

===In three dimensions===
An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, '''F'''<sub>1</sub>, '''F'''<sub>2</sub>, and '''F'''<sub>3</sub> that resist the weight of the object, <math>\mathbf{W} = -W\mathbf{\hat{k}}</math> (<math>\mathbf{\hat{k}}</math> is the unit vector in the vertical direction).  Let '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, and '''r'''<sub>3</sub> be the position coordinates of the support points, then the coordinates '''R''' of the center of mass satisfy the condition that the resultant torque is zero,
<math display="block">\mathbf{T} = (\mathbf{r}_1 - \mathbf{R}) \times \mathbf{F}_1 + (\mathbf{r}_2 - \mathbf{R}) \times \mathbf{F}_2 + (\mathbf{r}_3 - \mathbf{R}) \times \mathbf{F}_3 = 0,</math>
or
<math display="block">\mathbf{R} \times \left(-W\mathbf{\hat{k}}\right) = \mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times \mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3. </math>

This equation yields the coordinates of the center of mass '''R'''* in the horizontal plane as,
<math display="block"> \mathbf{R}^* = -\frac{1}{W} \mathbf{\hat{k}} \times (\mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times\mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3).</math>

The center of mass lies on the vertical line '''L''', given by
<math display="block"> \mathbf{L}(t) = \mathbf{R}^* + t\mathbf{\hat{k}}.</math>

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object.  The center of mass will be the intersection of the two lines '''L'''<sub>1</sub> and '''L'''<sub>2</sub> obtained from the two experiments.