In Newton's theory of gravitation, the Earth's gravitational field is given by, -GM r3 where 7 = (x, Y, z) is the position vector relative to the centre of the Earth, r = ||r||, M is the mass of the Earth and G is the gravitational constant. The force acting on a particle of mass m at position ř is given by F = mg(T). (a) A rocket of mass m is launched from position (0, 0, R) and travels to (0,0, R+ h) in a time T, along the straight line path C1. A similar rocket travels along the spiral path C2 given by h R+ 2nt = ñ(t): 2nt ,0, cos T sin with 0

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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5. In Newton's theory of gravitation, the Earth's gravitational field is given by,
-GM
g(F) =
p3
where 7 =
(x, y, z) is the position vector relative to the centre of the Earth, r =
|Fl, M is the mass of the
Earth and G is the gravitational constant. The force acting on a particle of mass m at position 7 is given by
F = mỹ(r).
(a) A rocket of mass m is launched from position (0, 0, R) and travels to (0, 0, R + h) in a time T, along the
straight line path C1. A similar rocket travels along the spiral path C2 given by
K() = (R+ ) (-
2nt
2nt
ř = h(t) :
- sin
-, 0, cos
T
T
with 0 <t<T. By evaluating the appropriate line integrals, show that the work done by the gravitational
field along C1 equals the work done along C2 (Figure 1).
(b) Show that the gravitational field is conservative by finding a potential d. Hence, verify the expression for
work obtained in 5a.
h
C2
Transcribed Image Text:5. In Newton's theory of gravitation, the Earth's gravitational field is given by, -GM g(F) = p3 where 7 = (x, y, z) is the position vector relative to the centre of the Earth, r = |Fl, M is the mass of the Earth and G is the gravitational constant. The force acting on a particle of mass m at position 7 is given by F = mỹ(r). (a) A rocket of mass m is launched from position (0, 0, R) and travels to (0, 0, R + h) in a time T, along the straight line path C1. A similar rocket travels along the spiral path C2 given by K() = (R+ ) (- 2nt 2nt ř = h(t) : - sin -, 0, cos T T with 0 <t<T. By evaluating the appropriate line integrals, show that the work done by the gravitational field along C1 equals the work done along C2 (Figure 1). (b) Show that the gravitational field is conservative by finding a potential d. Hence, verify the expression for work obtained in 5a. h C2
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