Recall that page VII promised ideas before names. It also promised a deductive approach. Let's apply this to the discussion of translational motion at the beginning of chapter 11 on page 255.
The second sentence says ``all the particles in the object move along identical parallel trajectories.
Figure 11.1 says all points on an object follow identical trajectories.
The first problem is, these terms are undefined and imprecise. Terminology including identical and parallel is introduced without prior explanation, and indeed without any posterior explanation. The meanings are not the least bit obvious; by way of counterexample consider that the geographic 42nd parallel is not "parallel" in the sense required here. Neither parallel nor trajectory appears in the chapter glossary.
The second problem is, even if it were clear and correct, it would be proof by bold assertion. It is more thaumaturgy than deduction.
Suggestion:
1) There is a super-easy way to explain and quantify translation in a way that uses and reinforces concepts that have already been introduced.
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For any two points (a) and (b), the displacement vector from (a) to (b) is independent of time. We can express this mathematically as
Xb(t) - Xa(t) = f(a,b) for all a, b, and t [1]
where the RHS depends on which two points we are talking about, (a) and (b), but is independent of time. The two positions Xa(t) and Xb(t) can change in some arbitrarily complicated way as a function of time, but the separation vector from one to the other is unchanging. This is what we call _translation_ or more specifically _translational motion_.
Secondly, we can differentiate equation [1] with respect to time to obtain
Vb(t) = Va(t) for all a, b, and t [2]
where each V is a velocity vector. It takes some attention to detail to obtain equation [2] rigorously, but it can be done. It means that at any given time, all points on the object partake of the same velocity. Recall that by definition, a vector has a magnitude and direction, but not a position. Point (a) has a different location from point (b), but the velocity Va and the velocity Vb are the exact same vector (at any given time).
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2) I see no need to mention the term «trajectory» at all ... certainly not «identical trajectories» or «parallel trajectories».
3) This is leads to a clear statement of what we mean by rigid-body motion:
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Consider motion where the distance between any two points is constant:
| Xb(t) - Xa(t) | = f(a,b) for all a, b, and t [3]
This is analogous to equation [1], but it says only that the scalar distance is constant (not the displacement vector). Technically speaking, that defines _isometry_. We call it _rigid-body motion_ if the positions are a continuous function of time, and there is isometry.
Obvious rigid-body motions include translations and rotations. (Mathematicians will tell you that in Euclidean geometry there are other isometries, including mirror reflection, but they are not continuous transformations, so we don't count them as rigid-body motions. What's worse, physicists will tell you that mirror reflection is _not_ an exact symmetry of the world we live in. If you build a clock using mirror-image blueprints, it won't necessarily work the same as the original clock. But let's not worry about that right now.)
There's a famous theorem that says every rotation in the plane leaves one point fixed. We call this point the _center_ of the rotation.
It turns out that at any given moment, the combination of a rotation plus a translation is just a rotation about some other center. (Homework: prove this.) Therefore there are only three cases we need to worry about, for rigid-body motion in the plane, at any given moment:
All points fixed: Stationary. Neither rotation nor translation.
One point fixed: Rotation.
No points fixed: Translation.
This three-way classification can change from moment to moment, as the object moves.
In some sense, a translation is the same as a rotation about a center that is infinitely far away. So if you understand everything about rotations, you understand translations for free.
Recall that page VII promised
ideas before names. It also promised adeductiveapproach. Let's apply this to the discussion of translational motion at the beginning of chapter 11 on page 255.The second sentence says ``all the particles in the object move along identical parallel trajectories.
Figure 11.1 says
all points on an object follow identical trajectories.The first problem is, these terms are undefined and imprecise. Terminology including
identicalandparallelis introduced without prior explanation, and indeed without any posterior explanation. The meanings are not the least bit obvious; by way of counterexample consider that the geographic 42nd parallel is not "parallel" in the sense required here. Neitherparallelnortrajectoryappears in the chapter glossary.The second problem is, even if it were clear and correct, it would be proof by bold assertion. It is more thaumaturgy than deduction.
Suggestion:
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For any two points (a) and (b), the displacement vector from (a) to (b) is independent of time. We can express this mathematically as
where the RHS depends on which two points we are talking about, (a) and (b), but is independent of time. The two positions Xa(t) and Xb(t) can change in some arbitrarily complicated way as a function of time, but the separation vector from one to the other is unchanging. This is what we call _translation_ or more specifically _translational motion_.
Secondly, we can differentiate equation [1] with respect to time to obtain
where each V is a velocity vector. It takes some attention to detail to obtain equation [2] rigorously, but it can be done. It means that at any given time, all points on the object partake of the same velocity. Recall that by definition, a vector has a magnitude and direction, but not a position. Point (a) has a different location from point (b), but the velocity Va and the velocity Vb are the exact same vector (at any given time).
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Consider motion where the distance between any two points is constant:
This is analogous to equation [1], but it says only that the scalar distance is constant (not the displacement vector). Technically speaking, that defines _isometry_. We call it _rigid-body motion_ if the positions are a continuous function of time, and there is isometry.
Obvious rigid-body motions include translations and rotations. (Mathematicians will tell you that in Euclidean geometry there are other isometries, including mirror reflection, but they are not continuous transformations, so we don't count them as rigid-body motions. What's worse, physicists will tell you that mirror reflection is _not_ an exact symmetry of the world we live in. If you build a clock using mirror-image blueprints, it won't necessarily work the same as the original clock. But let's not worry about that right now.)
There's a famous theorem that says every rotation in the plane leaves one point fixed. We call this point the _center_ of the rotation.
It turns out that at any given moment, the combination of a rotation plus a translation is just a rotation about some other center. (Homework: prove this.) Therefore there are only three cases we need to worry about, for rigid-body motion in the plane, at any given moment:
This three-way classification can change from moment to moment, as the object moves.
In some sense, a translation is the same as a rotation about a center that is infinitely far away. So if you understand everything about rotations, you understand translations for free.
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