Problems on central force

Question

Problems:

Problems on central force

(a) Suppose that the force on a particle of unit mass is

                                    F = F (r)e_r,

where e_r = r/r, with r = |r|. You may assume, without proof, that the velocity and acceleration of the particle are respectively given by

                  dr/dt = dr/dte_r + r dΘ/dt e_Θ

                   d²r/dt² = { d²r/dt² -r (dΘ/dt)²}er + (d²r/dt²+ 2 dr/dt dΘ/dt)eΘ

where er = cosΘ1+ sinΘj and eΘ= -sinΘ1+cosΘj. Show that

                            d²r/dt² - h²/r² = F (r),

where h is a constant. Further, using the substitution r = 1/u show that

                                du/dΘ = - 1/h dr/dt,

and that the path of the particle under the force F above is given by

                                     d²u/dΘ²+ u = - F(1/u)/h²u².

(b) A particle of unit mass is projected with speed √ 3µ/2a at right angles to the radius vector 2a at a distance a from a fixed point 0 and is subjected to a force F(r) = Thalr2, with p a positive constant. Show that the particle moves on a path which passes the centre at a distance of 3a at its maximum.