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:

- 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!) - 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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