Question

Please explain clearly. I need to know each steps reasons.

5) (Kepler's Problem) Suppose that a particle is moving in three dimensions under the influence of the force k F=- where k is a positive constant. (a) Find the torque acting on the particle with respect to the origin. Is angular momentum conserved? Show that the magnitude of the angular momentum is given by l = mr2ė. (b) Using Newton's second law, show that the momentum of the particle is given by p = c+ (mk/l) eg, where c is a constant vector. Then, show that the momentum vector moves along a circle. What is the radius and center of this circle? C) Using the definition of angular momentum and part(a), show that r = 12 mk(1 + € cos 0) cl where e = .. What does this equation represent geometrically? Discuss the mk special cases for e = 0,0<e < 1,6 = 1, ande > 1. Note that e, and e, are the unit vectors in polar coordinates

Answer 1

a) Ek @y To Torque , Z = řxt = rêrx r Hence torque is 2000 do dt ė So the angular momentum is conserved. moment of inestia angular momentum 1= I'w & angular speed I= mr and w= => [l= mrze met de er = mx éo lo => p = 8 + mk i Integrating wort-time . d. K V2 er dt d8 dt mu l Const. welare 1P-8] So momentum vector moves alonga circle Radius of the circle is mk e and center is at

l= my mr² for nozero l -> dt mr do d² mrw dv dr VE- r at ² 2 > mdr dv dr dt2 mr3 we have २ = put d at do mr2 and r= tu man havad), v(a)=-ku d?u m +U -- doz MK & 2 Solutions where ( 19 € (050] f where to cl mk mk (1 t60058) to eco Corresponds to circle el Corresponds hypere kola ECI Corresponds to ellipse E=1 Corresponds to pasabola.