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KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY INSTITUT DES SCIENCES ET TECHNOLOGIE Avenue de l'Armée, B.P. 3900 Kigali, Rwanda INSTITUTE EXAMINATIONS – ACADEMIC YEAR 2012-2013 FACULTY OF SCIENCE END OF SEMESTER EXAMINATION SECOND YEAR SEMESTER II MAT 3226 CLASSICAL MECHANICS TIME : 2 HOURS MAX.MARKS =60 DATE: ….. /…../2013 INSTRUCTIONS: 1. This Question Paper has 4 Questions in two sections; 2. The section A is compulsory and carries 30 marks 3. Answer to any 2 Questions in section B showing all working clearly. 4. Start every new question from a fresh page 5. Every question carries 15 marks in section B 6. Do not forget to write your Registration Number 7. Write all your answers in the booklet provided 8. Do not write any answers on this question paper. Section A: Q1. i. ii. iii. Verify the conservation laws for a simple harmonic oscillator. (6 marks) In the system with constraints, prove that the work done by the constraint force in any possible displacement is zero. (Hint: Consider that the mass is moving on a smooth fixed surface f ( x, y, z ) 0 in three dimension space). (6 marks) Show that the Lagrange’s equation of motion of a particle defined by L(q, q , t ) iv. v. 1 mq 2 V (q, t ) is equivalent to the Newton’s equation if q is a cartesian 2 coordinate. (6 marks) Considering a motion with a Lagrangian which does not depend explicitly on the time, establish the Jacobi’s integral of the motion. (6 marks) Verify if the transformation (q, p) (Q, P) produced by the generating function F2 ( P, q) P tan q is canonical transformation or not. (6 marks) Section B: Answer to any 2 questions in this section . Q2. Find the Lagrangian and Hamiltonian for a motion of a bead of mass m sliding smoothly on a wire in the shape z f (x) , the z –axis and x -axis are respectively vertical and horizontal (Make a sketch). (15 marks) Q3. Using the Newton’s second law, deduce that for a single particle with constant mass, the equation of motion implies the differential equation dT F .v where T stands dt for the kinetic energy, F the force and v the velocity. Moreover, show that if the mass varies with the time the corresponding equation becomes d (mT ) F . p where p is the dt linear momentum. (15 marks) Q4. A single particle moves in space under a conservative potential. Establish the Jacobi’s integral of the motion in ellipsoidal coordinates u , v, defined in terms of the usual cylindrical coordinates r , z , by the relations r a sinh v sin u z a cosh v cos u (15 marks)