Classical Dynamics: Question Sheet

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1 Pt 1B Advanced Physics Lent 5 Classical Dynamics: Question Sheet J. Ellis Questions are graded A to C in increasing order of difficulty. Energy Method 1(B) A ladder of length l rests against a wall at and angle θ to the vertical. There is no friction between the top of the ladder and the wall and no friction between the bottom of the ladder and the ground. Write down the kinetic and potential energies of the ladder as a function of θ and θ. (hint to work out the velocity of the centre of mass, write down expressions for the height and horizontal positions of the centre of mass as a function of θ and then differentiate with respect to time). Using the energy method outlined in lectures, derive the equation of motion in terms of θ that describes the way the ladder motion of the ladder. (B) A small solid cylinder of radius r lies inside a large tube of internal radius R. Find the period of oscillation of the small cylinder as it rocks too and fro about its equilibrium position. (hint show that the angular velocity of the small tube ω is related toθ, the angle between the centre of mass of the solid cylinder, the centre of the large tube and the vertical R r by ω = θ ) r Lagrangian Mechanics 3(A) Give, with suitable explanation, the number of degrees of freedom for the following dynamical systems - example on handout 1 (the double pendulum) ( Why the answer is not 6 three for each particle) - an insect crawling over the surface of a toy balloon. - the Earth/Sun/Moon system. - a diatomic molecule with a fixed interatomic distance. 4(A) Use the Lagrangian and the Euler-Lagrange equations for the ladder in question 1 to derive the equation of motion, and evaluate the momentum conjugate to θ. 1

2 5(B) A conical pendulum consists of a point mass m suspended by a string of length l so that it can swing freely below the point of suspension. If θ is the angle that the string makes to the vertical and φ is the azimuthal angle of the bob as seen from above, measured with respect to an arbitrary reference direction, write down the kinetic and potential energies of the system in terms of l, m, g θ and φ, and hence write down the Lagrangian. Using the Euler-Lagrange equation for the φ coordinate to show that ( m φ l θ ) sin is a constant of the motion and explain why this is so. From the Euler-Lagrange equation for the θ coordinate, show that the condition for the pendulum to make conical rotations i.e. ones with constant θ and φ, is: g φ = l cosθ θ φ m 6(B) A bead of mass m runs freely without friction along a circular wire which has a radius R and is mounted with the circle in a vertical plane. The circle is rotated about a vertical axis that passes through the centre of the circle at a constant angular velocity ω. Write down the Lagrangian for the system in terms of the angle θ between the downward vertical the centre of the circle of wire and the bead, and use the Euler Lagrange equation for the system to deduce the equation of motion of the bead. (note the similarity between this system and the conical pendulum of question 5). Find the Hamiltonian, H of for the system using the expression given in lectures. Is H a constant of the system? Is the energy of the system T+V a constant? Non-Inertial Frames 7(A) Paraboloidal telescope mirrors can be made by `spin casting', which involves rotating the molten glass and its container about a vertical axis as the glass solidifies. By considering the equilibrium of an element of the molten surface show that dy ω = x dx g where y is the height of the surface, x is the distance from the axis of rotation and ω is the angular velocity of rotation. For a mirror of focal length m, what angular velocity is required? [Ans: 1.57 radians s -1 ] 8(B) A train at latitude λ in the northern hemisphere is moving due north with a speed v along a straight and level track. Which rail experiences the larger vertical force? Show that the ratio R of the vertical forces on the rails is given approximately by 8Ωvhsin λ R = 1+ ga where h is the height of the centre of mass of the train above the rails which are a distance a apart, g is the acceleration of free fall, and Ω is the angular velocity of the Earth's rotation.

3 Normal Modes 9(A) When a diatomic molecule is adsorbed onto the surface of a metal the frequency of its internal vibrational mode is changed. If we consider a horizontal surface with the axis of the molecule vertical, a simple model which might describe this phenomenon is as follows. The molecule consists of two point masses, m, separated by a light spring of spring constant k. The mass nearer to the metal is attached to a fixed point (the metal surface) with a light spring of constant K. The two springs are collinear and the motion of the masses is regarded as confined to the line of the springs. Find the normal modes and how their frequencies vary with K. Sketch the results and comment on their physical significance for large and small K. [Ans. m ω = k + K ± k + K ] 4 1(C) A simple model of a jet engine comprises three identical thin rigid discs mounted equidistant on a uniform light shaft. Describe the normal modes of oscillation of the system. A small object entering the engine produces an abrupt change Ω in the angular velocity ω of the first disk. Obtain an expression for the maximum resultant angle of twist of the shaft between the discs, given that the angular frequency of the lowest vibrational normal mode is Ω. [Ans: Ω - an upper bound from the difference between two nonharmonically related 3 Ω sine-waves] 11(B) A uniform rod of length a hangs vertically on the end of an inelastic string of length a, the string being attached to the upper end of the rod. What are the frequencies of the normal ω = 5 ± 19 g ] modes of oscillation in a vertical plane? [Ans: ( ) a 1(A) The ringing note produced by a tea cup when it is tapped on the rim with a spoon is liable to vary in pitch depending on whereabouts in relation to the handle the cup is tapped. Predict this variation. 13(B) A block of mass M can move along a smooth horizontal track. Hanging from the block is a mass m on a string of length l that is free to move in a vertical plane that includes the line of motion of the block. Find the frequency and displacement patterns of the normal modes of oscillation of the system firstly by spotting the normal modes of the system and then secondly by writing the Lagrangian for the system and solving the Euler-Lagrange equations. M l m 3

4 Orbits 14(A) A satellite travelling round the Earth in a circular orbit with centre O experiences, at the point P, a sudden impulse which deflects it into a new orbit. Sketch the orbits to be expected if the impulse acts: (a) in the direction of the motion of the satellite; (b) in the reverse direction; (c) outwards along the line OP. On each sketch show clearly, in relation to O and P, the points A and B at which the new orbit is furthest from and nearest to O. 15(B) Two stars of unequal mass are in circular orbit about one another. One day the more massive star suffers a spherically symmetrical loss of matter (it explodes). After explosion the masses of the stars are equal. Show that the binary system will be disrupted if α > 3, where α is the ratio of the original masses. 16(B) In an attempt to place a satellite into a geostationary orbit, the correct speed and radial distance are achieved but the direction of motion has an angular error θ in the plane of the orbit. Show that the maximum and minimum radii of the orbit are r ( 1± sinθ ) where r is the radius of the geostationary orbit. By a short burst of its booster while at maximum radius, the satellite is able to put itself into a circular orbit. If the initial error θ is 1 minute of arc, calculate the period of this new orbit. [Ans: 1 day + 38 seconds.] 17(B) A point mass m on the end of a light string of length l is free to swing as a conical pendulum. Show that, in terms of the (constant) angular momentum J of the bob about the vertical axis, the energy of the pendulum may be written as 1 E = V ( θ ) + ml θ where J V ( θ ) = mgl( 1 cosθ ) + ml sin θ is the effective potential that determines the motion in θ. Sketch the form of ( θ ) differentiating V twice with respect to θ, show V and by (a) that the bob can move steadily round a circle, with θ = θ and angular velocity Ω given by Ω g = l cosθ (b) that, if the pendulum is given a little extra energy without changing its angular momentum, θ oscillates about θ with an angular frequency ω given by ω = Ω + ( 1 3cos θ ) 4

5 Rigid body dynamics 18(A) A uniform disc of radius.1 m and mass.4 kg is rotating with angular velocity 1rad/s about an axis at 45degees to its plane through its centre of mass. What is (a) its angular momentum, and (b) its kinetic energy? [Assume the centre of mass is stationary. Ans: (.7,, 1.4)x1-3 kg m s 1 w.r.t. obvious axes; 3/4mJ.] 19(B) A uniform rectangular tile drops without spinning until its corners reach positions (,, ), (a,, ), (a, b, ), (, b, ), when it strikes the top of a vertical pole at a point very close to the (,, ) corner. Just before impact the velocity of the tile was (,, -u). Assuming that the tile does not break, and that the impact is elastic (i.e. the kinetic energy of the tile is conserved), find immediately after impact (a) the velocity of its centre; (b) the angular momentum about its centre; (c) its angular velocity; Show that the velocity of the corner at (,, ) becomes (,, +u) immediately after impact. (B) Discuss and derive the basic rules of free precession that a spacecraft engineer must use in designing satellites and their control systems. 1(C) A coin of radius a is spun on a perfectly rough table is such a way that its centre is stationary, while its axis precesses steadily about the vertical at a fixed inclination θ with angular velocity Ω. Show that ω = 3 and hence that 4g Ω =. a sinθ Show also that if θ is a small angle the head of the coin, viewed from above, appears 3 to rotate with angular velocity Ω ( 1 cosθ ) = g θ / a 5

M2A2 Problem Sheet 3 - Hamiltonian Mechanics

M2A2 Problem Sheet 3 - Hamiltonian Mechanics MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,