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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

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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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