Center of Gravity

Center of Mass by Integration (Rigid Objects with Shape)
Center of Mass by Integration (Rigid Objects with Shape)

Computer drawing of a variety of objects showing the center of gravity - CG. CG = sum of component weight times component distance divided by total weight.

The center of gravity is a geometric property of any object.
The center of gravity is the average location of the
weight
of an object. We can completely describe the
motion
of any object through space in terms of the
translation
of the center of gravity of the object from one place to another, and the
rotation
of the object about its center of gravity if it
is free to rotate. In flight,
rockets
rotate about their centers of gravity.

Determining the center of gravity is very important
for any flying object.
How do engineers determine the location of the center of
gravity for a rocket which they are designing?

In general, determining the center of gravity (cg) is a complicated
procedure because the mass (and weight) may not be uniformly distributed
throughout the object. The general case requires the use of calculus
which we will discuss at the bottom of this page.
If the mass is uniformly distributed, the problem is greatly simplified.
If the object has a line (or plane) of symmetry, the cg lies
on the line of symmetry. For a
solid block of uniform material, the center of gravity is simply
at the average location of the
physical dimensions. For a rectangular block, 50 X 20 X 10,
the center of gravity is at the point (25,10, 5) .
For a triangle of height h, the cg is at h/3, and for a semi-circle of radius
r, the cg is at (4*r/(3*pi)) where pi is ratio of the circumference of the
circle to the diameter. There are tables of the location of the center of gravity
for many simple shapes in math and science books. The tables were generated
by using the equation from calculus shown on the slide.

For a general shaped object, there is a simple mechanical way to
determine the center of gravity:

  1. If we just balance the object using
    a string or an edge, the point at which the object
    is balanced is the center of gravity. (Just like balancing a
    pencil on your finger!)
  2. Another, more complicated way, is a two step method shown on
    the slide. In Step 1, you hang the object from any
    point and you drop a weighted
    string from the same point. Draw a line on the object along the
    string. For Step 2, repeat the procedure from another point on the object
    You now have two lines drawn on the object which intersect.
    The center of gravity is the point where the lines intersect. This
    procedure works well for irregularly shaped objects that are hard
    to balance.

If the mass of the object is not uniformly distributed, we must use calculus
to determine center of gravity.
We will use the symbol S dw to denote the integration of a continuous
function with respect to weight. Then the center of gravity can be determined from:

cg * W = S x dw

where x is the distance from a reference line, dw is an
increment of weight, and
W is the total weight of the object.
To evaluate the right side, we have to determine how the weight varies
geometrically. From the
weight equation, we know that:

w = m * g

where m is the mass of the object, and g is the gravitational
constant. In turn, the mass m of any object is equal to the
density, rho,
of the object times the
volume, V:

m = rho * V

We can combine the last two equations:

w = g * rho * V

then

dw = g * rho * dV

dw = g * rho(x,y,z) * dx dy dz

If we have a functional form for the mass distribution, we can solve the
equation for the center of gravity:

cg * W = g * SSS x * rho(x,y,z) dx dy dz

where SSS indicates a triple integral over dx. dy. and dz.
If we don’t know the functional form of the mass distribution,
we can numerically integrate the equation using a spreadsheet.
Divide the distance into a number of small volume segments and
determining the average value of the weight/volume (density times gravity) over
that small segment. Taking the sum of the average value of the weight/volume
times the distance times the volume segment
divided by the weight will produce the center of gravity.

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