Question

3. (a) [6 pts.] Recall that gravitional force exerted on a mass m placed at (x, y, z) by a mass M placed on the origin is given by F = **** ř, where r = (x, y, z) is the position vector of mass m. A potential for this force field is given by f = 9 GMm .. “, i.e. Vf = Assuming M is the earth and GM ~ 4 x 1014 m's-2, use Fundamental Theorem of Line Integrals to compute the work W against earth's gravitional field to move a satellite of mass m = 600 kg along any path from an orbit of altitude 2000 km to an orbit of altitude 4000 km. (b) [6 pts.] Find a potential function f(x, y) for the conservative vector field F = (6x sin y – sin x, 3x2 cos y + 1).

Answer 1

5. (a) Fundamental theorem of line integral tells s Fodă Es vector vied dă differensial ut position vector c- contour We can write Fo i Io Potential : Sot ar = s far = f(b) f(a)

. Work done = GM m 4000x103 6 mm 2000x10² = L Grml 2x100 4x10°46001 a 2 28 4x10x600x1 4x100 W = - 6x1000 l (Arswer

(6) Given F = (6xsiny- sinx, 3 oc² cosy +1) Lete e potential function. we have relation Bar gradient of porential function gives back conservative vector field. P = Px to py & op a 6xsiny-sina 1 (6x siny - sinx x

Px = 3x² siny + cos x Similarly ( Py = 3x² cas y + 1 & Py = 5(3x² cosy +1 ) dy Py = 3x² siny + y P = < 3x² siny+ cosi, 3x² siny + y ) is we were asked to represent pas f(x,y) f(x,y) = 3x² sinyt loss, 3x² sinyty> Answert