9.6
The Figure shows a cubical box that has been constructed from uniform
metal plate of negligible thickness. The box is open at the top and has edge
length L = 40 cm. Find (a) the
x
coordinate, (b) the
y
coordinate, and
(c) the
z
coordinate of the center of mass of the box.
A homogeneous symmetric body has its center of
mass (com) at its center of symmetry. This means
that the plates making up each side have their
coms at their respective centers. We can then
calculate the com of the entire box as the com due to
five point masses (each the mass of a side) at the
center point of each plate. Now, formally,
com
i i
i
1
r
m r
M
=
∑
G
G
i
i
M
m
=
∑
where
If the mass of the entire box is designated M then the mass of each side is M/5.
The vector equation above is 3 separate eqns, one for each coordinate axis.

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For the x axis,
x
i xi
i
1
1
M
M
M
M
M
r
m r
[
(0)
(0.2m)
(0.2m)
(0.2m)
(0.4m)]
M
M
5
5
5
5
5
=
=
+
+
+
+
∑
x
3
1
r
[
(0.2m)
(0.4m)]
5
5
=
+
x
r
0.20 m
=
Note that by the symmetry of the box and the location of its center along the x
direction (at x = 0.20 m) we could have anticipated this result. We do this for the
y direction finding that,
y
r
0.20 m
=
For the z axis,
z
i zi
i
1
1
M
M
M
M
M
r
m r
[
(0)
(0.2m)
(0.2m)
(0.2m)
(0.2m)]
M
M
5
5
5
5
5
=
=
+
+
+
+
∑

z
4
r
[
(0.2m)]
5
=
z
4
r
[
(0.2m)]
5
=
z
r
0.16 m
=

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