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that is our argument.
What good is the dot product? Are there any cases in physics where we
need it? Yes, we need it all the time. Eor instance, in Chapter 4 the kinetic
energy was called 3m03, but ïŸ the object is moving in space it should be the
velocity squared in the z-direction, the -direction, and the z-direction, and so
the formula for kinetic energy according to vector analysis is
K.E. = šm(0- 0) = šm(02 + 0y + 9)). (11.22)
Energy does not have direction. Momentum has direction; it is a vector, and 1$
1s the mass times the velocity vector.
--- Trang 229 ---
Another example of a dot product is the work done by a force when something
1s pushed from one place to the other. We have not yet deined work, but 1 1s
equivalent to the energy change, the weights lifted, when a force #' acts through
a distance s:
Work = È'!- 3. (11.23)
Tt is sometimes very convenient to talk about the component of a vector in
a certain direction (say the vertical direction because that is the direction of
gravity). For such purposes, it is useful to invent what we call a uw2t 0ector in
the direction that we want to study. By a unit vector we mean one whose dot
product with itself is equal to unity. Let us call this unit vector ?; then 2 - s = 1.
'Then, if we want the component of some vector in the direction of 2, we see that
the dot product ø - ? will be acosØ, i.e., the component of ø in the direction
of 2. This is a nice way to get the component; in fact, it permits us to get øiÏ
the components and to write a rather amusing formula. Suppose that in a given
system of coordinates, z, , and z, we invent three vectors: 2, a unit vector In
the direction zø; 7, a unit vector in the direction ; and &, a unit vector in the
direction z. Note frst that 2-# = 1. What is z- 7? When ÿwo vectors are at right
angles, their dot product is zero. 'Phus
:‹2=0 77=1
z-k=0 J3-k=0 k-k=l (11.24)
Now with these definitions, any vector whatsoever can be written this way:
œ = đ„9 + du) + a„k. (11.25)
By this means we can go from the components of a vector to the vector itself.
This discussion of vectors is by no means complete. However, rather than try
to go more deeply into the subject now, we shall first learn to use in physical
situations some of the ideas so far discussed. 'Phen, when we have properly
mastered this basic material, we shall fñnd it easier to penetrate more deeply into
the subject without getting too confused. We shall later fnd that it is useful
to defñne another kind of produet of two vectors, called the vector product, and
written as œ x Ö. However, we shall undertake a discussion of such matters in a
later chapter.
--- Trang 230 ---
( her'rcforrsÉfcs ©@Ê Foree©e
12-1 What is a force?
Although it is interesting and worth while to study the physical laws simply
because they help us to understand and to use nature, one ought to sÈop every
onece in a while and think, “What do they really mean?” 'Phe meaning of any
statement is a subjJect that has interested and troubled philosophers from time
Immemorial, and the meaning of physical laws is even more interesting, because
1t is generally believed that these laws represent some kind of real knowledge.
The meaning of knowledge is a deep problem in philosophy, and ït is always
Iimportant to ask, “What does it mean?”
Let us ask, “What is the meaning of the physical laws of Newton, which we
write as ` =ma? What is the meaning of force, mass, and acceleration?” Well,
we can intuitively sense the meaning of mass, and we can đefine acceleration ïŸ
we know the meaning of position and time. We shall not discuss those meanings,
but shall concentrate on the new concept of ƒorce. 'Phe answer is equally simpIle:
“Ha body is accelerating, then there is a force on it.” That is what Newton's laws
say, so the most precise and beautiful defnition of force imaginable might simply
be to say that force is the mass of an object times the acceleration. 5uppose we
have a law which says that the conservation of momentum is valid if the sum
of all the external forces 1s zero; then the question arises, “What does it mean,
that the sum of all the external forces is zero?” A pleasant way to define that
statement would be: “When the total momentum is a constant, then the sum of
the external forces is zero.” There must be something wrong with that, because it
is Just not saying anything new. If we have discovered a fundamental law, which
asserts that the force is equal to the mass times the acceleration, and then defne
the force to be the mass times the acceleration, we have found out nothing. We
could also defñne force to mean that a moving object with no force acting on i§
continues to move with constant velocity in a straight line. If we then observe an
object not moving in a straight line with a constant velocity, we might say that
--- Trang 231 ---
there is a force on it. Now such things certainly cannot be the content of physics,
because they are defnitions going In a circle. The Newtonian statement above,
however, seems to be a most precise definition of force, and one that appeals to
the mathematician; nevertheless, it is completely useless, because no prediction
whatsoever can be made from a definition. One might sit in an armchair all
day long and deñne words at will, but to ñnd out what happens when two balls
push against each other, or when a weight is hung on a spring, is another matter
altogether, because the way the bodies behøaue 1s something completely outside
any choice of definitions.
For example, if we were to choose to say that an object left to itself keeps its
position and does not move, then when we see something drifting, we could say
that must be due to a “gorce”——a gorce is the rate of change of position. Now we
have a wonderful new law, everything stands still except when a gorce is acting.
You see, that would be analogous to the above definition of force, and it would
contain no information. 'The real content of Newton”s laws is this: that the force
1s supposed to have some ?mdependent properties, in addition to the law P — ma;
but the speczfc independent properties that the force has were not completely
described by Newton or by anybody else, and therefore the physical law ` = na is
an ineomplete law. It implies that if we study the mass times the acceleration and
call the product the force, i.e., 1 we study the characteristics of force as a program
of interest, then we shall fnd that forces have some simplicity; the law is a good
program for analyzing nature, it is a suggestion that the forces will be simple.
Now the first example of such forces was the complete law of gravitation,
which was given by Newton, and ín stating the law he answered the question,
“What is the force?” If there were nothing but gravitation, then the combination
of this law and the force law (second law of motion) would be a complete theory,
but there is mụch more than gravitation, and we want to use Newton's laws in
many different situations. 'Therefore in order to proceed we have to tell something