Question

Q 4. (a) A body of mass m is moving in two dimensions in a constant z plane. Consider a coordinate system that rotates with constant angular speed 1 about the z-axis. In a fixed coordinate system (in the constant z plane), define the plane polar coordinates (r,0) while defining (r, ) as the corresponding plane polar coordinates in the rotating system. (i) In terms of the coordinate system rotating with constant angular speed 1, write down the kinetic energy (4 marks) (ii) Write down the Lagrangian function L for the particle motion in the rotating coordinate system in the plane polar coordinates (r, v) and indicate if there are any cyclic coordinates. [3 marks) (iii) If no force acts on the particle, determine the equations of motion in the rotating coordinate system, in plane polar coordinates (r, v). (5 marks) (iv) Re-express the equations of motion you obtained in part (a)(iii) in terms of the fixed coordinate system (r,) and state what physical quantity these resultant equations imply is conserved. [3 marks) (b) Explain why it might be necessary to account for the rotation of the Earth in a given set of equations. On what order of scale would you expect the rotation of the Earth to have any effect. (3 marks) (c) In your own words, provide an overview of how the effect of the rotation of the Earth can be accounted for within the equations of motion for a particle of mass m moving at the surface of the Earth and briefly discuss how the ideas are applied to the Foucault pendulum example. (8 marks) [Total marks for question: 26 marks] END OF QUESTION 4 THE END OF THE COURSEWORK