Page 1 of 6 PHYS3032 Classical Mechanics Past Paper Write your answers on blank papers. Clearly indicate the question number at the beginning of each of your answers. You must show all your steps. No credits will be given if the final answer is correct but the steps are wrong. In contrast, partial credits will be given to correct steps toward the find answer. All units are given in SI units unless otherwise stated. Answer in term of the given parameters when numerical values are not given in the problems. Page 2 of 6 Question 1 (10 pt) A satellite is moving in a uniform circular orbit of radius about Earth. By what fraction must its velocity be increased for the satellite to be in an elliptical orbit with perigee min = and apogee max = 2 Question 2 (10 pt) The Lagrangian = 1 2 || 2 + ( ) yields the correct equation of motion of a particle with charge in static magnetic fields, = × where () is the vector potential of the static magnetic fields such that = × For a constant magnetic field = 0 , the vector potential is given by = 0 2 ( + ). (a) Rewrite the Lagrangian in cylindrical coordinate (, , ). (5 pt) (b) Besides the total energy, identify another conserved quantity of the system. Explain or prove explicitly why it is conserved. (5 pt) Page 3 of 6 Question 3 (15 pt) A smooth rod of length rotates in a horizontal plane with a constant angular velocity about an axis fixed at the end of the rod and perpendicular to the plane of rotation. A bead of mass is released initially from rest (relative to the rod) at a position 4 from the fixed end shown in the figure below. The bead is confined to move without friction along the rod only. Ignore gravity. Give answers in terms of the basis vectors and coordinates of the rotating frame. (a) Write down the equation of motion of the bead with respect to the rotating frame. (5pt) (b) Find the displacement of the bead along the rod, , as a function of time. (5pt) (c) Find the force acting on the bead by the rod as function of time. (5pt) Question 4 (20 pt) A three-particle system, in which the relative positions among the particles are fixed, consists of masses and the masses are fixed at body coordinates (1, 2, 3) as follows: 1 = 3, ( , 0, ) 2 = , (, , ) 3 = , (, , ) The system is rotating about along 3-axis at angular velocity . (a) Show the inertia tensor of the system with respect to the body coordinate system is = ( 7 0 1 0 10 0 1 0 7 ) 2. (5 pt) (b) Find the values of principal moments of inertia. (5 pt) (c) Find the torque acting on the system with respect to an inertial observer who shares the same origin of the body coordinate. (5 pt) (d) Find the basis vector (1 , 2 , 3 ) of a set of orthogonal principal axes. (5 pt) Fixed End , rotating with the rod Page 4 of 6 Question 5 (20 pt) Consider a solid cylinder with radius R and mass m being attached with a spring at its center. The spring has a spring constant and a natural length . The cylinder rolls along the incline without slipping. One of your goals is to find the static friction between the surface and the cylinder. The system is under uniform gravity, . (a) Define your choice of the generalized coordinates (draw a picture to specify your coordinates). (3 pt) (b) Write down the equation of constraint(s) associated to your generalized coordinates. (2 pt) (c) Use the method of Lagrange Multiplier to find the static friction between the surface of the incline and the cylinder. Leave the answer in terms of the coordinate and the parameters. Solving the coordinate as an explicit function of time is not required. (10 pt) (d) Find the oscillation frequency, , of the system. (5 pt) Express your answer in terms of the generalized coordinate and the parameters of the problem: , , , , and . R Rolling without slipping Moment of inertia about the symmetric axis through the center of mass: = 1 2 2 Page 5 of 6 Question 6 (23 pt) Consider the coupled system shown above. The system block attached to the spring, which has a spring constant , can move horizontally on the frictionless surface. The pendulum attached to the block has a fixed length and is subject to a uniform gravity, . The mass of the block and the pendulum are both . The mass of the string on the pendulum is negligible. The parameters satisfy 2 = . The system is at equilibrium when = 0 and = 0. (a) Show the Lagrangian function is (5pt) ( , , , ) = 2 + cos + 1 2 2 2 1 2 2 + cos In case of small oscillations in , (b) show the equation of motion of the system is (6 pt) ( 2 1 ) ( ) + ( 2/ 0 0 ) ( ) = ( 0 0 ) (c) find the characteristic frequencies (normal mode frequencies, ) in terms of and ; (4pt) (d) find eigenvectors of the two normal modes; (5pt) (e) write down the general solution of the system and identify the parameters to be determined by the initial conditions in the general solution. (3pt) k Page 6 of 6 Question 7 (12 pt) A thin uniform U-shaped tube is filled with a certain amount of water shown in the figure on the right. The length of the tube occupied by water is > . If one can neglect damping and the air pressure, what is the oscillation period of water in the tube The system is under a uniform gravity as shown in the figure. Assuming the oscillation is small such that the semi-circular part of the tube is always occupied by water. You can assume the cross-section of the tube is and the density of water is .
