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general rule that the area oŸ a zone of a sphere is proportional to its axial width.) |
'Therefore the potential energy due to đn is |
đW =— Gm đm " Gm'2maụu da |
But we see that |
r? =g2+(R—z)°=2++?+ R—2Ra |
=a?+R”—2R+. |
2rdr = —2Rd+z |
dy — dĩ |
--- Trang 266 --- |
'Therefore, |
Gm'2ma_u đr |
dW = P › |
and so , Rịca |
W- Gm2malu J dự |
t Tì+a |
Œmm'2mxaju 2 Œm(4ma?) |
= R TT R |
= R. (13.18) |
Thus, for a thin spherical shell, the potential energy of a mass ?m/, external to |
the shell, is the same as though the mass of the shell were concentrated at its |
center. The earth can be imagined as a series of spherical shells, each one of |
which contributes an energy which depends only on its mass and the distance |
from its center to the particle; adding them all together we get the £o‡œl mmass, |
and therefore the earth acts as though all the material were at the centerl |
But notice what happens if our point is on the ?ws¿de of the shell. Making |
the same calculation, but with ? on the inside, we still get the diference of the |
©wo r's, but now in the form a— jÈ— (œ-+ R) = —2, or minus twice the distance |
from the center. In other words, W comes out to be W = —Œmmrn/a, which is |
¿ndependen‡t of F and independent of position, ¡.e., the same energy no matter |
tohere we are inside. 'Pherefore no force; no work is done when we move about |
inside. Tf the potential energy is the same no matter where an object is placed |
inside the sphere, there can be no force on it. So there is no force inside, there 1s |
only a force outside, and the force outside is the same as though the mass were |
all at the center. |
--- Trang 267 --- |
MVor'k (ra ốổl IPo£ortfterl Froorggg, (c©ortecltrsrom) |
14-1 Work |
In the preceding chapter we have presented a great many new ideas and |
results that play a central role in physics. 'These ideas are so important that 1t |
seems worth while to devote a whole chapter to a closer examination of them. |
In the present chapter we shall not repeat the “proofs” or the specifc tricks by |
which the results were obtained, but shall concentrate instead upon a discussion |
of the ideas themselves. |
In learning any subject of a technical nature where mathematics plays a role, |
one is confronted with the task of understanding and storing away in the memory |
a huge body of facts and ideas, held together by certain relationships which can |
be “proved” or “shown” to exist between them. It is easy to confuse the proof |
1tself with the relationship which it establishes. Clearly, the important thing |
to learn and to remember is the relationship, not the proof. In any particular |
circumstance we can either say “it can be shown that” such and such is true, or |
we can show it. In almost all cases, the particular proof that is used is concocted, |
ñirst of all, in such form that it can be written quickly and easily on the chalkboard |
or on paper, and so that it will be as smooth-looking as possible. Consequently, |
the proof may look deceptively simple, when in fact, the author might have |
worked for hours trying diferent ways of calculating the same thing until he has |
found the neatest way, so as to be able to show that it can be shown in the |
shortest amount of timel 'The thing to be remembered, when seeing a proof, is |
not the proof itself, but rather that it can be shoun that such and such is true. |
Of course, if the proof involves some mmathematical procedures or “tricks” that |
one has not seen before, attention should be given not to the trick exactly, but |
to the mathematical idea. involved. |
Tt is certain that in all the demonstrations that are made in a course such |
as this, not one has been remembered from the time when the author studied |
--- Trang 268 --- |
freshman physics. Quite the contrary: he merely remembers that such and such |
1s true, and to explain how it can be shown he invents a demonstration at the |
mmoment ¡% is needed. Anyone who has really learned a subject should be able |
to follow a similar procedure, but it is no use remermbering the proofs. 'That is |
why, in this chapter, we shall avoid the proofs of the various statements made |
previously, and merely sumnmarize the results. |
The frst idea that has to be digested is t0ork dơne bụ a force. The physical |
word “work” is not the word in the ordinary sense of “Workers of the world |
unitel,” but is a different idea. Physical work is expressed as ƒ F': ds, called “the |
line integral of P' dot đs,” which means that if the force, for instance, is In one |
direction and the object on which the force is working is displaced in a certain |
direction, then omlu the component oƒ force ïn the dicction oƒ the displacement |
does any work. If, for instance, the force were constant and the displacement |
were a finite distance As, then the work done in moving the object through that |
distance is only the component of force along As times Az. The rule is “force |
times distance,” but we really mean only the component of force in the direction |
of the displacement tỉimes As or, equivalently, the component of displacement in |
the direction of force times #'. It is evident that no work whatsoever is done by |
a force which is at right angles to the displacement. |
Now 1ƒ the vector displacement As is resolved into components, in other |
words, if the actual displacement is As and we want %o consider i% efectively |
as a component of displacement Az in the z-direction, A# ïn the -direction, |
and Az in the z-direction, then the work done in carrying an object from one |
place to another can be calculated in three parts, by calculating the work done |
along z, along , and along z. The work done in goïing along # involves only that |
component of force, namely #„, and so on, so the work is F„ Az + tụ Au+ ty Az. |
'When the force is not constant, and we have a complicated curved motion, then |
we must resolve the path into a lot of little As”s, add the work done in carrying |
the object along each As, and take the limit as As goes to zero. Thịs is the |
meaning of the “line integral.” |
Everything we have just said is contained in the formula W = Ƒ#'- ds. It |
is all very well to say that it is a marvelous formula, but it is another thing to |
understand what it means, or what some of the consequences are. |
The word “work” in physics has a meaning so diferent from that of the word |
as it is used in ordinary circumstances that it must be observed carefully that |
there are some peculiar circumstances in which it appears not to be the same. |
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