2. Let us consider a particle of mass m moving in three-dimensional space. This particle is acted on by a force whose potential is given by U(r) k/r2. The particle is coming from infinity with an initial velocity V... (a) [10 pts] Let us define the contributions from the potential energy U(r) and the angular momentum part to the Hamiltonian collectively as is effective potential U(r). What effective potential for the radial motion of the particle when its angular momentum is L? Express the angular momentum L with the impact an parameter b. the (b) [15 pts] Depending on its initial condition, the particle undergoes one of the following two motions: 1) it approaches to a minimum distance min 0 and then goes away back to the infinite distance, or 2) it collides with the center (r=0) of the potential. Let us consider the second case of 2). What is the maximum impact parameter bmax for the particle to fall to the force center at T 0? Write down the total cross section for this falling-in process. 18 (c) [15 pts] Show that the particle moves in a logarithmic spiral orbit given by r=re for b<bmax, where is the azimuthal angle of the particle's position vector in the plane angular momentum vector. Express L, m, and k. normal to its in terms of