422
CHAPTER 44
44.83 Find the center of mass of the cube of edge a with three faces on the coordinate planes, if the density is numerically
equal to the sum of the distances from the coordinate planes.
The mass
The moment about the
xz-plane is
By symmetry,
So
44.84 Find the center of mass of the first octant of the ball of radius a, x
2 + y
2 + z
2 s a
2 , if the density is numerically
equal to z.
The mass is one-eighth of that of the ball in Problem 44.77, or ira 116. The moment about the xzplane is
Hence,
By symmetry,
The moment about the xy-plane is
Hence
Thus, the center of mass is
44.85 Find the center of mass of a solid right circular cone ^ of height h and radius of base b, if the density is equal to the
distance from the base.
Let the cone have the base
z = h — (hlb)r. By symmetry,
in the xy-plane and vertex at (0, 0, h); its equation is then
The mass
Use cylindrical coordinates. Then
The moment about the xyplane is
Thus, the center of mass is
Hence,
44.86
Find the moments of inertia of the triangle bounded by 3* + 4y = 24, x = 0, and y = 0, and having
density 1.
The moment of inertia with respect to the *-axis is
The moment of inertia with respect to the y-axis is
44.87 Find the moment of inertia of a square plate of side a with respect to a side, if the density is numerically equal to
the distance from an extremity of that side.
Let the square be
and let the density at (x, y) be the distance
from the
origin. We want to find the moment of inertia I x about the Jt-axis:
Now, by the
must be equal
symmetry of the situation, the moment of inertia about the y-axis,
to I x . This allows us to write