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that in this case, the force is changing as we go along, it is not just a constant.
As we know, the force is —GŒM/rẺ tỉmes the mass ?m, where ?m is the mass that
moves. Now certainly when a body falls toward the earth, the kinetic energy
Increases as the distance fallen increases, just as it does when we do not wOorry
about the variation of force with height. The question is whether it is possible to
ñnd another formula for potential energy diferent from rmgh, a diferent function
of distance away from the earth, so that conservation of energy will still be true.
This one-dimensional case is easy to treat because we know that the change
in the kinetic energy is equal to the integral, from one end of the motion to the
other, of —ŒMmn/r2 times the displacement đr:
1;—7¡== | GMm —>- (13.11)
'There are no cosines needed for this case because the force and the displacement
are in the same direction. It is easy to integrate dr/z2; the result is —1/z, so
Eq. (13.11) becomes
Tạ — Tì =+GMm[ - ¬) (13.12)
T2 TỊ
Thus we have a diferent formula for potential energy. Equation (13.12) tells us
that the quantity ($zmø2 — GMm/r) calculated at poïnt 1, at poïnt 2, or at any
other place, has a constant value.
W©e now have the formula for the potential energy in a gravitational fñeld for
vertical motion. NÑow we have an interesting problem. Can we make perpetual
tmotion in a gravitational fñeld? "The gravitational field varies; in diferent places
1t is in diferent directions and has diferent strengths. Could we do something
like this, using a fxed, frictionless track: start at some point and lift an object
out to some other point, then move it around an arc to a third point, then lower
1t a certain distance, then move it in at a certain slope and pull it out some other
way, so that when we bring it back to the starting point, a certain amount of work
--- Trang 257 ---
has been done by the gravitational force, and the kinetic energy of the object
is increased? Can we design the curve so that it comes back moving a little bit
faster than it did before, so that it goes around and around and around, and gives
us perpetual motion? Since perpetual motion is impossible, we ought to fnd
out that this is also impossible. We ought to discover the following proposition:
since there is no friction the object should come back with neither higher nor
lower velocity——it should be able to keep going around and around any closed
path. 5tated in another way, he totaÏ tuork đone ín goïng arouwnd a complete
cụcÌe should be zero for gravity forces, because ïÍ it is not zero we can get energy
out by going around. (Tf the work turns out to be less than zero, so that we
get less speed when we go around one way, then we merely go around the other
way, because the forces, of course, depend only upon the position, not upon the
direction; if one way is plus, the other way would be minus, so unless it is zero
we will get perpetual motion by goỉng around either way.)
° : 6
M3 4
Fig. 13-3. A closed path ¡in a gravitational field.
ls the work really zero? Let us try to demonstrate that it is. First we shall
explain more or less why it is zero, and then we shall examine it a little better
mathematically. Suppose that we use a simple path such as that shown In
Fig. 13-3, in which a small mass is carried from point 1 to point 2, and then is
made to go around a circle to 3, back to 4, then to 5, 6, 7, and 8, and fñnally
back to 1. AlI of the lines are either purely radial or circular, with ă as the
center. How much work is done in carrying m around this path? Between points
1 and 2, it is GŒMm tìmes the difference of 1/r between these bwo points:
Wha =Í Esds= | -GMm =GMm( = ^)
1 1 r T2 TỊ
tHrom 2 to 3 the force is exactly at right angles to the curve, so that W2¿ = 0.
The work from 3 to 4 is
Mai = ƒ E-ds= GAm( TC — n):
3 T4 T3
--- Trang 258 ---
In the same fashion, we find that Was = 0, Wss = GMm(1/re — 1/rs), Wsy =0,
W7s = GMm(1/rs — 1/r;), and Wsy =0. Thus
1 1 1 1 1 1 1 1
W=GMm( + TT tam}
T2 — T1 PA Tạ T6 T5 T§ã Tĩ
But we note that ra = 73, 74 — 7s, re =r7;, and rs =r\. Therefore W =0.
°
lo x Llb
Fig. 13-4. A “smooth” closed path, showing a magnified segment of
It approximated by a series of radial and circumferential steps, and an
enlarged view of one step.
Of course we may wonder whether this is too trivial a curve. What iÝ we use
a real curve? Let us try i9 on a real curve. First of all, we might like to assert
that a real curve could always be imitated sufficiently well by a series of sawtooth
Jiggles like those of Fig. 13-4, and that therefore, etc., Q.E.D., but without a
little analysis, it is not obvious at first that the work done going around even a
small triangle is zero. Let us magnify one of the triangles, as shown in EFig. 13-4.
1s the work done in going from ø to b and ö to c on a triangle the same as the
work done in going directly from a to c? Suppose that the force is acting in a
certain direction; let us take the triangle such that the side be is in this direction,
Just as an example. We also suppose that the triangle is so small that the force
1s essentially constant over the entire triangle. What is the work done in goïng
from ø to c? lt is
W. = J E'-ds = Fscos0,
since the force is constant. Now let us calculate the work done in going around the
other ©wo sides of the triangle. Ôn the vertical side œb the force is perpendicular
--- Trang 259 ---
to đs, so that here the work is zero. Ôn the horizontal side be,
MS F'-ds = Ea.
Thus we see that the work done in going along the sides of a small triangle is
the same as that done going on a slant, because scosØ is equal to ø. We have
proved previously that the answer is zero for any path composed of a series of
notches like those of Fig. 13-3, and also that we do the same work iŸ we cut across
the corners instead oŸ going along the notches (so long as the notches are ñne
enough, and we can always make them very fine); therefore, Éhe Uork done ïn
goïng around œnụ path ímn a grauitatlional field ts zero.
'This 1s a very remarkable result. It tells us something we did not previousÌy
know about planetary motion. It tells us that when a planet moves around the
sun (without any other objects around, no other forces) it moves in such a manner
that the square of the speed at any point minus some constants divided by the