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The concept of gravitational energy in the context of planetary motion. It discusses how the total energy of a planet in circular or elliptical orbit around a center of force is given by eq. 14-25, which includes both kinetic and potential energy. The document also explains how the total energy determines the semimajor axis and period of the orbit, and how changing the speed of an orbiting satellite requires a change in radius. A sample problem is provided to illustrate the concepts.
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v::: V2,L ::: It,,r ~
U1~ tr-- d (1 r )
wf-- IT'f. ~ 'c-u U ~ f.t--~ (Lr)
v""-- ') C L r- ) ~ ~ kr IJ 7 +-- ~J (I t-rCA-..(y")
~ (?-! 'I ~ {f~/- ~(9 4-;7(y/ tr1~)
2 ~ i ~(9 -I- !-I- ~
1 -:. ~ <A... r9- G---t2 :: r (y.. ~ «0. l c)
r
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l
/ )^ J.",^ tty )^ ~ .¢ e..j^ 1.^ /Ij
if,,"1--": Y>(5r) v
L
FIGURE 14-19. Sample Problem 14-10. The orbits of space
noncircular orbit at lower height above the Earth. The relative size of the Earth and the orbital heights is not to scale.
After the burn, the speed decreases by the given amount of 0.95%
The potential energy of B at point P immediately after the short
be E' K' + U' = 9.58 X 1010 J + 2(- 9.76 X 1010 J)
= ......... 2(-9.94 X lOiO J)
6.52 X 106 m 6520 km, a reduction of 1.8% from the value in the original orbit. The corre sponding period is
417'2(6.52 X 106 mi ) == ((6.67 X 10- 11 N' m^2 /kg^2 )(5.98 X 10 H^ kg)
'" 5240 s.
rockets) at t = 105 s, then A returns to P at t 5380 s (deter
tial passage, or at t = 105 s + 5240 s 5345 s. Thus B is now
identical in strength and duration to the first but in the reverse di rection. This returns B to the original circular orbit, now 35 s
1 4-8 THE GRAVITATIONAL FIELD (Optional)
A basic fact of gravitation is that two particles exert forces on one another. We can think of this as a direct interaction between the two particles, if we wish. This point of view is called action-at-a-distance, the particles interacting even though they are not in contact. Another point of view is the field concept, which regards a particle as modifying the space around it in some way and setting up a gravitational field. This field, the strength of which depends on the mass of the particle, then acts on any other particle, exerting the force of gravitational attraction on it. The field therefore plays an intermediate role in our thinking about the force that one particle exerts on another. According to this view we have two separate parts to our problem. First, we must determine the gravitational ficld established by a given distribution of particles. Sec ond, we must calculate the gravitational force that this field exerts on another particle placed in it. We use this same approach later in the text when we study electromagnetism, in which case particles with elec tric charge set up an electric field, and the force on another charged particle is determined by the strength of the electric field at the location of the particle. Let us consider the Earth as an isolated particle and ig nore all rotational and other nongravitational effects (so that g and go are equivalent). We use a small test body of mass
placed in the vicinity of the Earth, it will experience a force having a definite direction and magnitude at each point in space. The direction is radially in toward the center of the Earth, and the magnitude is mog. We can associate with each point near the Earth a vector g, which is the accelera tion that a body would experienee if it were released at this point. We define the gravitational field strength at a point as the gravitational force per unit mass at that point or, in terms of our test mass,
F g= (14-26)
By moving the test mass to various positions, we can make a map showing the gravitational field at any point in space. We can then find the force on a particle at any point in that field by multiplying the mass m of the particle by the value of the gravitational field g at that point: F mg. Figure 14-20 shows examples of gravitational fields. The gravitational field is an example of a vector field, each point in this field having a vector associated with it. There are also scalar fields, such as the temperature field in a heat-conducting solid. The gravitational field arising from
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