F33OT1-E1 The University of Nottingham SCHOOL OF PHYSICS AND ASTRONOMY A LEVEL 2 MODULE, AUTUMN SEMESTER EXAM 2011–2012 PRINCIPLES OF DYNAMICS Time allowed ONE Hour THIRTY Minutes Candidates may complete the front cover of their answer book and sign their desk card but must NOT write anything else until the start of the examination period is announced. Answer THREE out of FIVE Questions Only silent, self-contained calculators with a Single-line Display or Dual-line Display are permitted in this examination. Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English provided that neither language is the subject of this examination. Subject specific translation dictionaries are not permitted. No electronic devices capable of storing and retrieving text, including electronic dictionaries, may be used. An indication is given of the approximate weighting of each part of a question by means of a bold figure enclosed by curly brackets, e.g. {2}, immediately following that part. DO NOT turn examination paper over until instructed to do so. F33OT1-E1 Turn over 2 F33OT1-E1 You must answer 3 out of 5 questions. You should aim to spend about 30 minutes on each question. 1 The motion of a particle of mass m and position r(t) moving under the influence of a potential V (r) is described by a Lagrangian L = 12 m|˙r|2 − V (r), where r = |r|. (a) Show that the equation of motion is given by m¨r = −V ′ (r)ˆ r, where ˆr = rr {6}. [Hint: You may wish to use the fact that the gradient of the potential, ∇V (r) = V ′ (r)ˆ r. ] (b) (c) (d) Show that h = r × r˙ is conserved, and explain the physical significance of this result {4}. Using the result of part (b), show that the motion of the particle must lie in a plane {3}. Using polar coordinates (r, θ) in the plane of motion, show that the Lagrangian can be written as 1 L = m(r˙ 2 + r2 θ˙2 ) − V (r) 2 ˙ r + rθ˙ˆθ, and write down the equations of motion {6}. [Hint: you may assume that r˙ = rˆ where ˆr and ˆθ are the unit vectors along the r and θ directions respectively. ] (e) Verify that the following quantities are conserved ˙ |h| = r2 θ, 1 E = mr˙ 2 + Veff (r) 2 where Veff (r) = V (r) + 12 m |h| , and explain the physical significance in each case {6}. r2 2 F33OT1-E1 3 2 (a) F33OT1-E1 A rigid body with angular velocity w is modelled by a system of particles of mass, mk located at position x(k) (t), relative to a fixed point. (i) Show that the angular momentum of the kth particle is h(k) = mk x(k) × (w × x(k) ) {5} and hence show that the total angular momentum of the rigid body, ∑ ) ( {3} h= mk |x(k) |2 w − (x(k) · w)x(k) k (ii) [Hint: You may use the vector identity a × (b × c) = b(a · c) − c(a · b).] Using this result, show that in the continuum limit, the moment of inertia tensor has components, ∫ [ ] Iij = ρ(x) (|x|2 δij − xi xj dV {8} body (b) A model of an ice skater is shown in figure 2.1. The body is crudely modelled as a uniform cuboid of mass M − m, and the arms are modelled as a uniform rod of mass m and length l. The moment of inertia tensors for the cuboid and the rod are given by  2    1 d + h2 0 0 0 0 2 1 1 0 d2 + h2 0  , Irod = ml2  0 12 0  Icuboid = (M − m)  12 12 0 0 2d2 0 0 1 (i) (ii) Find the full moment of inertia tensor of the skater {2}. Use this model to explain why the skater finds it easier √ to pirouette more quickly when she tucks her arms in (you should assume that l > 2d){7}. Figure 2.1: Skater with arms outstretched, modelled as a cuboid with a rod passing through its centre. F33OT1-E1 Turn over 4 3 (a) F33OT1-E1 A system is described by generalised coordinates, qa and a Lagrangian, L = L(qa , q˙a ) with no explicit time dependence. Show that ∑ H= pa q˙a − L a (b) is conserved and give the physical meaning of this result. {7} In General Relativity, the motion of a planet of mass, m, in the gravitational field of the Sun can be described by generalised coordinates, (r(t), θ(t), ϕ(t)), and a Lagrangian L = −mcs where s= √ c2 f (r) − r˙ 2 − r2 (θ˙2 + ϕ˙ 2 sin2 θ), f (r) f (r) = 1 − rs r and rs = 2GM⊙ /c2 is known as the Schwarzschild radius of the Sun. (i) Show that the generalised momenta are given by pr = (ii) (iii) mcr˙ , sf pθ = mcr2 θ˙ , s pϕ = mcr2 ϕ˙ sin2 θ s {6} Identify the ignorable coordinate, and give the physical meaning of the corresponding conserved quantity. {3} Show that the conserved energy of the planet is given by E= mc3 f s {6} What is the energy of a stationary planet, infinitely far away from the Sun? {3} F33OT1-E1 5 4 (a) (b) F33OT1-E1 State Hamilton’s principle for a system described by generalised coordinates, qn (t), where n = 1, . . . N , and derive the Euler-Lagrange equations. {12} A system of N particles, each of mass, m, and position (xn , yn (t)), where the x-components are fixed, is described by a Lagrangian L= N ∑ 1 n=1 (i) 2 my˙n2 − N −1 ∑ n=1 1 N k(yn+1 − yn )2 2 Show that for n = 2, . . . , N − 1, the corresponding equations of motion are given by m¨ yn = −N k(2yn − yn−1 − yn+1 ) (ii) (iii) By writing xn = {4} (4.1) n N and yn (t) = y(xn , t), show that ( ) 1 ′′ 1 2yn − yn−1 − yn+1 ≈ − 2 y (xn , t) + O N N3 where ′ denotes partial differentiation with respect to xn {4}.( [Hint: ) you may use the following result: yn±1 = yn ± N1 y ′ (xn , t) + 2N1 2 y ′′ (xn , t) + O N13 ] Show that in the limit N → ∞, m → 0, holding the total mass, M = mN , and the constant, k fixed, equation (4.1) implies that M y¨(x, t) = ky ′′ (x, t) where we have dropped the index on the x-component for brevity. {5} F33OT1-E1 Turn over 6 5 F33OT1-E1 Figure 5.1 shows a rod of mass m and length l attached to a well oiled pivot, A, that is constrained to move along a wire, given by y = f (x). The rod is assumed to be swinging freely under the influence of gravity, in the xy plane. At time t the rod makes an angle θ(t) with the downward vertical, and x(t) denotes the horizontal position of the pivot A. (a) Show that the Lagrangian describing the system is given by ( ) 1 1 2 ˙2 1 l ′2 2 ′ ˙ L = m(1+f )x˙ + ml θ + mlx˙ θ(cos θ−f sin θ)+mg f (x) + cos θ 2 6 2 2 (b) (c) {13} [Hint: you may assume that the moment of inertia of the rod about its centre is given by 1 I = 12 ml2 ] Now consider small oscillations about the equilibrium point at x = x∗ , θ = 0, where f ′ (x∗ ) = 0. Show that the Lagrangian may be approximated by ( ) 1 2 1 2 ˙2 1 1 l ′′ 2 2 L = mϵ˙ + ml θ + mlϵ˙θ˙ + mg f (x∗ )ϵ − θ + constant 2 6 2 2 4 where ϵ = x − x∗ . {6} Hence show that the equations describing these small oscillations can be written ( of motion ) ϵ ¨ = −M ψ where ψ = as K ψ and lθ ( ) ( ) g −f ′′ (x∗ )l 0 1 12 K= , M= {6} 1 1 1 0 l 2 2 3 Figure 5.1 Rod of mass m and length l attached to a pivot on the wire given by y = f (x). F33OT1-E1 End