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* Linear motion refers to motion in space without regard to rotation; angular motion specifically refers to the rotation of a body about any axis (the body may or may not be undergoing linear motion at the same time).
Mass, Center of Mass, and Moment of Inertia j 5
know or be capable of calculating these mass properties. Letís look at a few definitions first.
In general, people think of mass as a measure of the amount of nlatter in a body For our purposes in the study of mechanics, we can also think of mass as a measure of a bodyís resistance to motion or a change in its motion. Thus, the greater a bodyís mass, the harder it will be to set it in motion or change its motion.
In laymenís terms, the center of mass (also known as center of gravity) is the point in a body around which the mass of the body is evenly distributed. In mechanics, the center of mass is the point through which any force can act on the body without resulting in a rotation of the body
Although most people are familiar with the terms mass and center of gravity, the term moment of inertia is not so familiar; however, in mechanics it is equally important. The mass moment of inertia of a body is a quantitative measure of the radial distribution of the mass of a body about a given axis of rotation. Analogous to mass being a measure of a bodyís resistance to linear motion, mass moment of inertia is a measure of a bodyís resistance to rotational motion.
Now that you know what these properties mean, letís look at how to calculate each.
For a given body made up of a number of particles, the total mass of the body is simply the sum of the masses of all elemental particles making up the body, where the mass of each elemental particle is its mass density times its volume. Assuming that the body is of uniform density, then the total mass of the body is simply the density of the body times the total volume of the body This is expressed in the following equation:
p dV = p J dV
In practice, you rarely need to take the volume integral to find the mass of a body especially considering that many of the bodies we will consider, for example, cars and planes, are not of uniform density You will simplify these complicated bodies by breaking them down into an ensemble of component bodies of known or easily calculable mass and simply sum the masses of all components to arrive at the total mass.
The calculation of the center of gravity of a body is a little more involved. First, divide the body into an infinite number of elemental masses with the center of each mass specified relative to the reference coordinate system axes. Next, take the first moment of each mass about the reference axes and then add up all of these moments. The first moment is the product of the mass times the distance along a given coordinate axis from the origin to the center of mass. Finally divide this sum of moments by the total mass of the body, yielding the coordinates to the center of mass of the body relative to the reference axes. You must perform this calculation once for each dimension, that is, twice when working in 2D and three times when working in 3D. Flere are the equations
6 j Chapter 1: Bask Concepts
for the 3D coordinates of the center of mass of a body:
where (x, y, z)c are the coordinates of the center of mass for the body and (x, y, 2)0 are the coordinates to the center of mass of each elemental mass. The quantities xD dm, y0 dm, and z0 dm represent the first moments of the elemental mass, dm, about each of the coordinate axes.
Here again, donít worry too much about the integrals in these equations. In practice you will be summing finite numbers of masses, and the formulas will take on the friendlier forms shown here:
Note that you can easily substitute weights for masses in these formulas since the constant acceleration due to gravity, g, would appear in both the numerators and denominators, thus dropping out of the equations. Recall that the weight of an object is its
where mt is the total mass, m; is the mass of each point mass in the system, CG is the combined center of gravity, and eg; is the location of the center of gravity of each point mass in design, or reference, coordinates. Notice that CG and cgj are shown as vectors, since they denote position in Cartesian coordinates. This is a matter of convenience, since it allows you to take care of the x, y, and z (or just x and y in two dimensions) components in one shot.
In the code samples that follow, letís assume that the point masses making up the body are represented by an array of structures in which each structure contains the point
mass times the acceleration due to gravity, g, which is 32.174 ft/s2 (9.8 m/s2) at sea level.
The formulas for calculating the total mass and center of gravity for a system of discrete point masses can conveniently be written in vector notation as follows:
CG = [^(cgjXmi)] jmt
Mass, Center of Mass, and Moment of Inertia | 7
massís design coordinates and mass. The structure will also contain an element to hold the coordinates of the point mass relative to the combined center of gravity of the rigid body, which will be calculated later. -y