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It turns out that there is one particularly natural and useful way to make
a clear definition, but it requires a brief digression. Every object has a
balance point, referred to in physics as the center of mass. For a two-
dimensional object such as a cardboard cutout, the center of mass is the
point at which you could hang the object from a string and make it balance.
In the case of the ballerina (who is likely to be three-dimensional unless her
diet is particularly severe), it might be a point either inside or outside her
body, depending on how she holds her arms. Even if it is not practical to
attach a string to the balance point itself, the center of mass can be defined
as shown in the figure on the left.
Why is the center of mass concept relevant to the question of classifying
rotational motion as opposed to motion through space. As illustrated in
the figure above, it turns out that the motion of an object’s center of mass is
nearly always far simpler than the motion of any other part of the object.
The ballerina’s body is a large object with a complex shape. We might
expect that her motion would be much more complicated that the motion
of a small, simply-shaped object, say a marble, thrown up at the same angle
as the angle at which she leapt. But it turns out that the motion of the
ballerina’s center of mass is exactly the same as the motion of the marble.
That is, the motion of the center of mass is the same as the motion the
ballerina would have if all her mass was concentrated at a point. By restrict-
ing our attention to the motion of the center of mass, we can therefore
simplify things greatly.
No matter what point you hang the
pear from, the string lines up with the
pear’s center of mass. The center of
mass can therefore be defined as the
intersection of all the lines made by
hanging the pear in this way. Note that
the X in the figure should not be
interpreted as implying that the center
of mass is on the surface — it is
actually inside the pear.
We can now replace the ambiguous idea of “motion as a whole through
space” with the more useful and better defined concept of “center-of-mass
motion.” The motion of any rigid body can be cleanly split into rotation
and center-of-mass motion. By this definition, the tipping chair does have
both rotational and center-of-mass motion. Concentrating on the center of
center of mass
The leaping dancer’s motion is
complicated, but the motion of her
center of mass is simple.
The same leaping dancer, viewed from
above. Her center of mass traces a
straight line, but a point away from her
center of mass, such as her elbow,
traces the much more complicated
path shown by the dots.
Section 2.1Types of Motion
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