Physical Applications of Triple Integrals
Mass and Static Moments of a Solid
Suppose we have a solid occupying a region U. Its volume density at a point M (x, y, z) is given by the function ρ (x, y, z). Then the mass of the solid m is expressed by the triple integral as
The static moments of the solid about the coordinate planes
The coordinates of the center of gravity of the solid are described by the expressions:
If a solid is homogeneous with density
Moments of Inertia of a Solid
The moments of inertia of a solid about the coordinate planes
and the moments of inertia of a solid about the coordinate axes
As it can be seen, the following properties are valid:
The moment of inertia about the origin is called the integral
The moment of inertia about the origin can be expressed through the moments of inertia about the coordinate planes as follows:
Tensor of Inertia
Using the
This tensor is symmetric and, hence, it can be transformed to a diagonal view by choosing the appropriate coordinate axes
If a body rotates about an axis which does not coincide with a principal axis of inertia, it will experience vibrations at the high rotation speeds. Therefore, when designing such devices it is necessary the axis of rotation to be coinciding with one of the principal axes of inertia. For example, when replacing car tires, it's often necessary to balance the wheels by attaching small lead weights to ensure the coincidence of the rotation axis with the principal axis of inertia and to eliminate vibration.
Gravitational Potential and Attraction Force
The Newton potential of a body at a point
where
The integration is performed over the whole volume of the body. Knowing the potential, one can calculate the force of attraction of the material point of mass
where
Solved Problems
Example 1.
Find the centroid of a homogeneous half-ball of radius
Solution.
We introduce the system of coordinates in such a way that the half-ball is located at
Using this system of coordinates, we find the centroid (the center of gravity) of the solid. Obviously, by symmetry,
Calculate the coordinate
Since the half-ball is homogeneous, we set
The symbol
It remains to compute the triple integral
Thus, the coordinate
Example 2.
Determine the mass and coordinates of the center of gravity of the unit cube with the density
(Figure
Solution.
First we calculate the mass of the cube:
Now we calculate the static moments
Similarly, we find the moments
Calculate the coordinates of the center of gravity of the cube: