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centers of mass.
Center-of-mass motion in one dimension
In addition to restricting our study of motion to center-of-mass motion,
we will begin by considering only cases in which the center of mass moves
along a straight line. This will include cases such as objects falling straight
down, or a car that speeds up and slows down but does not turn.
Note that even though we are not explicitly studying the more complex
aspects of motion, we can still analyze the center-of-mass motion while
ignoring other types of motion that might be occurring simultaneously .
For instance, if a cat is falling out of a tree and is initially upside-down, it
goes through a series of contortions that bring its feet under it. This is
definitely not an example of rigid-body motion, but we can still analyze the
motion of the cat’s center of mass just as we would for a dropping rock.
Self-Check
Consider a person running, a person pedaling on a bicycle, a person coasting
on a bicycle, and a person coasting on ice skates. In which cases is the
center-of-mass motion one-dimensional. Which cases are examples of rigid-
body motion.
2.2Describing Distance and Time
Center-of-mass motion in one dimension is particularly easy to deal
with because all the information about it can be encapsulated in two
variables: x, the position of the center of mass relative to the origin, and t,
which measures a point in time. For instance, if someone supplied you with
a sufficiently detailed table of x and t values, you would know pretty much
all there was to know about the motion of the object’s center of mass.
A point in time as opposed to duration
In ordinary speech, we use the word “time” in two different senses,
which are to be distinguished in physics. It can be used, as in “a short time”
or “our time here on earth,” to mean a length or duration of time, or it can
be used to indicate a clock reading, as in “I didn’t know what time it was,”
or “now’s the time.” In symbols, t is ordinarily used to mean a point in
time, while
.
t signifies an interval or duration in time. The capital Greek
letter delta,
.
, means “the change in...,” i.e. a duration in time is the change
or difference between one clock reading and another. The notation
.
t does
not signify the product of two numbers,
.
and t, but rather one single
number,
.
t. If a matinee begins at a point in time t=1 o’clock and ends at
t=3 o’clock, the duration of the movie was the change in t,
.
t = 3 hours - 1 hour = 2 hours .
To avoid the use of negative numbers for
.
t, we write the clock reading
“after” to the left of the minus sign, and the clock reading “before” to the
right of the minus sign. A more specific definition of the delta notation is
therefore that delta stands for “after minus before.”
Even though our definition of the delta notation guarantees that
.
t is
positive, there is no reason why t can’t be negative. If t could not be nega-
tive, what would have happened one second before t=0. That doesn’t mean
Coasting on a bike and coasting on skates give one-dimensional center-of-mass motion, but running and pedaling
require moving body parts up and down, which makes the center of mass move up and down. The only example of
rigid-body motion is coasting on skates. (Coasting on a bike is not rigid-body motion, because the wheels twist.)
Section 2.2Describing Distance and Time