Consider a point x = to of stable equilibrium. show that in the vicinity of the point to, the potential energy function is equal (up to an arbitrary constant) to the potential energy of a harmonic oscillator with harmonic constant k = v(20). hint: taylor expand the potential energy function about the point x = lo and keep up to quadratic terms. (b) consider the central force + = (ab sin (be) – mon) ê, with (a, b) positive con- stants, m the mass of the particle and p the polar, radial coordinate on the plane. i. [2 marks] find a potential energy function ii. [3 marks] find the effective potential vefs() from the energy conservation equation. iii. (5 marksconsider a particle moving on a plane with total angular momen- tum per unit mass h = b and total energy e = i. qualitatively sketch the energy diagram (clearly denote the turning points and distinguish the regions where motion is and is not possible). iv. (5 marks) if the particle is found moving in the vicinity of a point of stable equilibrium, denote the apses of its orbit on the energy diagram and find its velocity when it passes by each apsis. assume that the 'maximum/minimum distance pmin/max from the centre of force is known.