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10.2 Center of Mass 319

(a) (b)
2x
O
x O
x
y Here y is the distance
below the vertex. We sum slabs of
thickness dy and
2
area (2x) .
x

The large triangle is a
vertical cross section
y
through the pyramid.
y
FIGURE 10.16 (a) Cross section through the pyramid. The triangle in blue shows that
at a height y measured from the apex, the half-width of the pyramid is x y tan . (b) The
thin horizontal slab indicated in red is a square measuring 2x 2x with a thickness dy.

Thus the mass of the slab of thickness dy at this height y is
2 2 2
dm dV (2y tan ) dy 4 (tan )y dy
Equation (10.26) then gives us the y coordinate of the center of mass:

1 1 2
y y dm 4r(tan f)y dy (10.29)
3
CM
M M
The total mass is
2 2
M dm 4r(tan f)y dy (10.30)

2
When we substitute Eq. (10.30) into Eq. (10.29), the common factor 4 tan
cancels, leaving
y dy
3
y (10.31)
y dy
CM
2
As we sum the square slabs of thickness dy in both of these integrals, the integra-
tion runs from y 0 at the top of the pyramid to y h at the bottom, where h is
the height of the pyramid. Evaluation of these integrals yields

h 3 y 4 h h 4
y dy `
0 4 0 4
h 2 y 3 h h 3
y dy `
0 3 0 3
The y coordinate of the center of mass is therefore

4
h /4 3
y CM 3 h
h /3 4
This means that the center of mass is 3/4 147 m below the apex; that is, it is 1/4
147 m 37 m above the ground.