BSc/MSci EXAMINATION PHY-304 Time Allowed: Physical Dynamics 2 hours 30 minutes Date: 28 th May 2009 Time: 10:00 Instructions: Answer ALL questions in section A. Answer ONLY TWO questions from section B. Section A carries 40 marks, each question in section B carries 30 marks. An indicative marking-scheme is shown in square brackets [ ] after each part of a question. COMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND CROSS THROUGH ANY WORK WHICH IS NOT TO BE ASSESSED. NUMERIC CALCULATORS ARE PERMITTED IN THIS EX- AMINATION. Data: A formula sheet containing mathematical results that may be of help in various questions is provided at the end of the examination paper. Examiners: Dr G Travaglini (CO) Prof WJ Spence (DCO) YOU ARE NOT PERMITTED TO START READING THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR c Queen Mary, University of London 2009
SECTION A: Answer ALL questions in this section A1. Consider a system of n particles of masses m i, i =1,...,n which are subject to external and internal forces; let F (e) i be the external force acting on particle i, and F ji the internal force due to particle j on particle i. (i) Define what is the coordinate R of the centre of mass. [3] (ii) Show that the motion of the centre of mass of the system is the same as that of a pointlike particle of mass M := n i=1 m i subject to a total force equal to n i=1 F (e) i. Assume that the weak form of the action and reaction principle holds, i.e. that F ij = F ji. [4] (iii) Show that the total kinetic energy of the system can be rewritten as T = 1 2 M Ṙ2 + T, where T is the relative kinetic energy. [6] A2. Consider the three-dimensional system described by the Lagrangian L = m 2 (ẋ2 +ẏ 2 +ż 2 ) V (x, y, z). In each of the following cases, discuss what is the most general form of the potential V such that the physical quantities listed are conserved, and explain what are the Noether symmetries responsible for their conservation. (i) The ŷ-component of the momentum, p y. [3] (ii) The three components of the angular momentum of the particle, L. [3] (iii) The ẑ-component of the angular momentum of the particle, L z. In this case the potential is also such that the L x and L y components are not conserved. [4] Page 1 of 7 Question continues over page
A3. Consider the motion of a particle with one degree of freedom parameterised by a coordinate q, in a potential V (q). The mass of the particle is m. (i) Define the action of the particle in terms of the Lagrangian L(q, q), and state the principle of least action. [4] (ii) Using the principle of least action, derive the Lagrange equations for the particle. [6] (iii) Let E be the total energy of the system. Explain why no motion is possible for E<V(q). [3] (iv) Show that the energy E(q, q) := (1/2) m q 2 + V (q) is conserved if V (q) is independent of time. [4] Page 2 of 7
SECTION B: Answer ONLY TWO QUESTIONS from this section B1. A particle of mass m is free to slide on a plane without friction, and attached to one end of a light string of length l. The other end of the string is passed through a small, frictionless hole O and is attached to a mass M that hangs below the hole and can move only vertically. We also assume that the string remains taut at all times. Gravity acts as usual in the vertical direction. (i) How many degrees of freedom does the system have? [3] (ii) Choosing an appropriate set of generalised coordinates, write down the Lagrangian of the system and the Lagrange equations. [8] (iii) Determine the conserved quantities and write down expressions for them. What are the Noether symmetries associated to their conservation? [6] (iv) Use the Lagrange equations to prove that there is a solution of the motion where the particle m moves at a constant distance R from the hole O, while the mass M is at rest. Determine what is the angular velocity of the mass m for this case, as well as the value of the conserved quantities for the system. [7] (v) Calculate the frequency of the small oscillations around the solution discussed in point (iv) as a function of m, M, the constant distance R, and the gravity acceleration g. [6] Page 3 of 7
B2. A pointlike particle of mass m 1 is constrained to move on a circle of radius R 1 and centre at the origin O of an inertial system (ˆx, ŷ, ẑ). The circle is contained in the (ˆx, ŷ) plane. A second pointlike particle of mass m 2 is constrained to move on a second circle, with radius R 2 and centre at the point (0, 0,h); this second circle is contained in the plane z = h. A spring of elastic constant k connects the two particles, so that the system has a potential V =(1/2) kd 2, where d is the distance between the two particles (see Figure below). The system is not subject to gravity. (i) How many degrees of freedom does the system have? [3] (ii) Choosing appropriate generalised coordinates, write down the Lagrangian and the Lagrange equations. [8] (iii) Explain what are the Noether symmetries of the problem, and write down the explicit expressions for the corresponding conserved quantities in terms of the generalised coordinates. Interpret the physical meaning of these conserved quantities. [6] (iv) Find a stable equilibrium position for the system, and determine the frequency of the small oscillations around this equilibrium position. [9] (v) The second circle is now displaced so that its new centre is at point (a, b, h), where a and b are arbitrary constants. The second circle is still contained in the plane z = h. Without performing any calculations, discuss the symmetries of the problem in this new setup; from this, determine which of the conserved quantities found in the setup of part (iii) above (i.e. when the centre of the second circle was at (0, 0,h)) are still conserved, and which are not. [4] Page 4 of 7
B3. Consider a square made of four identical rigid rods of length l and mass m. The mass per unit length of each rod is constant. The square can move in the vertical plane (ˆx, ŷ), with one of its corners fixed at point O, so that the square can oscillate freely in the vertical plane about the point O. Gravity acts as usual in the vertical direction. (i) How many degrees of freedom does the system have? [3] (ii) Calculate the moment of inertia of the square with respect to an axis orthogonal to the (ˆx, ŷ) plane and passing through the centre of mass of the square. Use the parallel axis theorem to calculate the moment of inertia of the square with respect to an axis orthogonal to the (ˆx, ŷ) plane and passing through the fixed point O. [8] (iii) Choosing appropriate generalised coordinates, write down the Lagrangian of the system and the Lagrange equations. [7] (iv) Determine the frequency of small oscillations around the equilibrium position. [3] (v) Consider now the case where the square has a constant mass per unit area σ equal to σ =4m/l 2. Calculate the new frequency of small oscillations about the stable equilibrium position. Will the system oscillate more rapidly in this situation, or in the situation considered previously where the mass of the square is concentrated along the edges, with constant mass per unit length? [9] Page 5 of 7
B4. Consider the motion of a single particle of mass m subject to a generic potential V (r), where r := (x, y, z) is the position vector of the particle in an inertial system (ˆx, ŷ, ẑ) with origin at O. (i) Write down the Hamiltonian H of the system and the Hamilton equations. [4] (ii) Given a physical observable O(r, p), prove that the time evolution of O is described by the equation Ȯ = {O,H}, where {A, B} := A B x n p n A p n B x n is the Poisson bracket of A with B (a sum over n is understood), and p is the momentum of the particle. [6] (iii) Prove that the vector A defined by A := p L mk r r, and the angular momentum L are orthogonal. k denotes a positive constant. [4] (iv) Consider now the case where V is a central potential, V = V (r), where r := r. Explain why the orbits are planar, and why the vector A is parallel to the plane containing the orbit. [6] (v) Consider now the case where V is the gravitational potential V = k/r. Using the result in part (ii) above, prove that A is a conserved quantity, i.e. calculate explicitly the Poisson bracket {A,H} and show that {A,H} = 0. [10] Page 6 of 7
FORMULA SHEET Plane polar coordinates: r := r ˆr, ṙ = ṙ ˆr + r φ ˆφ, ṙ 2 = ṙ 2 + r 2 φ2. Cylindrical coordinates: r := r ˆr, ṙ = ρ ˆρ + ρ φ ˆφ + ż ẑ, ṙ 2 = ρ 2 + ρ 2 φ2 + ż 2. Vector algebra identity: a (b c) =b (a c) c (a b). End of Examination Paper Dr G Travaglini c Queen Mary, University of London 2009 Page 7 of 7