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moving on to the second slice, estimating its volume and adding that to the volume of
the first slice; and then moving on to the third, and fourth, and so on, aggregating the
volume of the loaf as you move toward the other end. Integration applies this technique
to infinitely thin slices of volume to compute the volume of any arbitrary shape. The
same techniques apply to other computations—for example, computing areas, iner‐
tias , masses, and so on, and even aggregating distance traveled over successive small
slices of time, as you'll see later. In fact, this latter application is the inverse of the de‐
rivative of distance with respect to time, which gives speed. Using integration and dif‐
ferentiation in this way allows you to work back and forth when computing speed,
acceleration, and distance traveled, as you'll see shortly. In fact, we'll use these concepts
heavily throughout the rest of this topic.
Mass, Center of Mass, and Moment of Inertia
The properties of a body— mass , center of mass , and moment of inertia , collectively called
mass properties— are absolutely crucial to the study of mechanics, as the linear and
angular 1 motion of a body and a body's response to a given force are functions of these
mass properties. Thus, in order to accurately model a body in motion, you need to know
or be capable of calculating these mass properties. Let's look at a few definitions first.
In general, people think of mass as a measure of the amount of matter in a body. For
our purposes in the study of mechanics, we can also think of mass as a measure of a
body's resistance to motion or a change in its motion. Thus, the greater a body's mass,
the harder it will be to set it in motion or change its motion.
In laymen's 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 body's resistance to linear motion, mass moment of inertia (also known as rotational
inertia ) is a measure of a body's resistance to rotational motion.
Now that you know what these properties mean, let's 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
1. Linear motion refers to motion in space without regard to rotation; angular motion refers specifically to the
rotation of a body about any axis (the body may or may not be undergoing linear motion at the same time).
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