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Hà. — F,.
Now we can fnd out how the torque transforms by merely substituting for
+, , and z” the expressions (20.3), and for F2, f;„, f;/ those given by (20.4),
all into (20.2). 5o, we have a rather long string of terms for 7x⁄„ and (rather
surprisingly at fñrst) it turns out that it comes right down to #„ — F„, which
--- Trang 365 ---
we recognize to be the torque in the z-plane:
Tạ: = (œcos 8 + sin Ø)(F„ cos Ø — F„ sỉn 6)
— (cos Ø — #zsin Ø)(F„ cos Ø + Ƒ+„ sin 0)
= #F,(cos? Ø + sin? Ø) — F„(sin? Ø + cos? Ø)
+ #ƑF+(— sin Ø cos Ø - sin Ø cos 8)
+ #(sin Ø cos Ø — sin Ø cos 0)
= ky — UF„ = Tụy. (20.5)
'That result is clear, for iŸ we only turn our axes ?n fhe pÏøœne, the twist around z
in that plane is no diferent than it was before, because it is the same planel
What will be more interesting is the expression for 7„;:, because that is a new
plane. We now do exactly the same thing with the z/z-plane, and it comes out
as follows:
Tựụ/z = (ucosØ — zsin Ø)F„
— Z(E„ cos Ø — EF„ sin 6)
= (0E; — zFu) cosØ + (zF„ — œF,) sin 0
= Tụz„ COs Ở -E 7„„ sỉn . (20.6)
Einally, we do it for Z4”:
Tz„: = Z(F„ cos 9 + Fy sin 8)
— (# cos 8 + sin 0);
= (zF„ — #F,) cos Ø — (WEF+„ — zF) sin 9
= Tz„ cOs Ở — Tựz sỉn Ö. (20.7)
W©e wanted to get a rule for fñnding torques in new axes in terms oŸ torques In
old axes, and now we have the rule. How can we ever remember that rule? lf we
look carefully at (20.5), (20.6), and (20.7), we see that there is a close relationship
between these equations and the equations for #z, , and z. IÝ, somehow, we could
call 7„„ the z-componen# of something, let us call it the z-component of 7, then it
would be all right; we would understand (20.5) as a vector transformation, since
the z-component would be unchanged, as it should be. Likewise, if we associate
with the z-plane the #ø-component of our newly invented vector, and with the
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zz-plane, the -component, then these transformation expressions would read
Tự: — Tự,
T„/ = T„ cOS Ö + Tụ sỉn Ø, (20.8)
Tụ: = Tụ COS Ở — 7„ sỉn Ø,
which is just the rule for vectorsl
Therefore we have proved that we may identify the combination of #2, — 1F»
with what we ordinarily call the z-component of a certain artificially invented
vector. Although a torque is a twist on a plane, and it has no ø ør?or¿ vector
character, mathematically it does behave like a vector. 'This vector is at right
angles to the plane of the twist, and its length is proportional to the strength
of the twist. The three components of such a quantity will transform like a real
VCCfEOF.
So we represent torques by vectors; with each plane on which the torque is
supposed to be acting, we associate a line at right angles, by a rule. But “at
right angles” leaves the sign unspecifed. 'To get the sign right, we must adopt a
rule which will tell us that if the torque were in a certain sense on the z-plane,
then the axis that we want to associate with it is in the “up” z-direction. 'Phat is,
somebody has to defne “right” and “left” for us. Supposing that the coordinate
system is øz, , z In a ripght-hand system, then the rule wïll be the following: 1f
we think of the twist as If we were turning a screw having a right-hand thread,
then the direction of the vector that we will associate with that bwist is in the
direction that the screw would advanee.
'Why is torque a vector? It is a miracle of good luck that we can associate a
single axis with a plane, and therefore that we can associate a vector with the
torque; it is a special property of three-dimensional space. In two dimensions, the
torque is an ordinary scalar, and there need be no direction associated with it. In
three dimensions, it is a vector. If we had four dimensions, we would be in great
difficulty, because (ïf we had time, for exarmple, as the fourth dimension) we would
not only have planes like #ø, z, and zz, we would also have #z-, #-, and z-
planes. There would be s#z of them, and one cannot represent six quantities as
one vector in four dimensions.
WSe will be living in three dimensions for a long time, so iÈ is well to notice
that the foregoing mathematical treatment did not depend upon the fact that +
was position and # was force; it only depended on the transformation laws for
vectors. Therefore If, instead of z, we used the ø-component of some other vector,
--- Trang 367 ---
1E is not going to make any difference. In other words, iŸ we were to calculate
„bu — aub„, where œ and b are vectors, and call it the z-component of some new
quantity c, then these new quantities form a vector c. We need a mathematical
notation for the relationship of the new vector, with i0s three components, to
the vectors œ and b. The notation that has been devised for this is e = œ x b.
W© have then, in addition to the ordinary scalar produect in the theory of vector
analysis, a new kind of product, called the øector product. Thus, 1Í œ— œ x b,
this is the same as writing
C„ = quÖ; — dzb„,
đụ = „by — d„Ù„, (20.9)
cy = q„bu — dub„.
TỶ we reverse the order of ø and b, calling œ, b and b, œø, we would have the sign of e
reversed, because c„ would be b„ø„ — b„a„. Therefore the cross product is unlike
ordinary multiplication, where øÖ = ba; for the cross product, bx œ=— —ø x b.
tErom this, we can prove at once that if œ = b, the cross product 1s zero. Thus,
øxqœ=0.
'The cross product is very important for representing the features of rotation,
and it is important that we understand the geometrical relationship of the three
vectors ø, b, and e. Of course the relationship in components is given in Eq. (20.9)
and from that one can determine what the relationship is in geometry. “The
answer is, frst, that the vector e is perpendicular to both œ and b. (Try to
calculate e - œ, and see if it does not reduce to zero.) Second, the magnitude of e
turns out to be the magnitude oŸ ø times the magnitude of b times the sine of
the angle between the two. In which direction does e point? Imagine that we
turn ø into b through an angle less than 180”; a screw with a right-hand thread
turning in this way will advance in the direction of e. The fact that we say a
righi-hand screw instead of a /eff-hand screw is a convention, and is a perpetual
reminder that if ø and b are “honest” vectors in the ordinary sense, the new kind