PHYS2006 Classical Mechanics 23/04/2018

10 Classical Mechanics - exercise sheet 10

ur final set of exercises addresses the normal modes of coupled oscillators. Please post your solutions into the box opposite the First Year Labs by 2pm on Wednesday 9th May.

Reading Read about coupled oscillators and normal modes in your favourite textbook. e.g.

Fowles & Cassiday Analytical Mechanics (7th ed.) section 11.3 Chow Classical Mechanics (2nd ed.) sections 11.1-11.2 Kibble & Berkshire Classical Mechanics (5th ed.) sections 11.1-11.4 Thornton & Marion Classical Dynamics (5th ed.) chapter 12

1 Symmetry and invariance (5 marks) a) State the principle of Time Translational Invariance and show with an example how this gives rise to a

complex exponential form for the oscillatory solution. b) State the physical quantity whose conservation results from this time translational invariance. c) With reference to the quantum mechanical wavefunction for a freely-moving particle, show how the

conservation rule and oscillatory solution are consistent.

2 Crystal vibrations (5 marks) A one-dimensional crystal may be modelled as comprising a series of N beads of mass m, connected by springs of constant c, so that, at rest, the beads are spaced equally a distance a apart and the first and last beads are connected by similar springs to fixed points a further distance a beyond these beads. When the 1-D crystal undergoes longitudinal oscillations, the displacement of the nth bead from its rest position is given by un, where u0 and uN+1 may hence refer to the fixed points at the ends of the crystal. a) Show that the equations of motion are, for n = 1, …, N,

( )112

2

2d

d−+ +−= nnn

n uuumc

tu

.

b) Substitute a normal mode solution of the form un = An e-iωt = A einϑ e-iωt and show that ω and ϑ are related by

=

2sin4 22 ϑω

mc

.

3 Suspended masses (5 marks) A system of masses and springs is shown in the adjacent figure, for which k1 = 3 N m-1, k2 = 7.5 N m-1, and k3 = 39 N m-1. Assume that the blocks move vertically with no rotation of side-to-side swinging. a) Show that the normal mode angular frequencies are 2 rad s-1 and 3 rad s-1, and find

the corresponding normal modes. The blocks are held in their equilibrium positions; the 2 kg block is then displaced 1 cm upwards and the system is released from rest. b) Find expressions for the subsequent displacements of both blocks.

4 Coupled pendulums (5 marks) Two identical masses m are suspended by light strings of length l. The suspension points are a distance L apart and a light spring of natural length L and spring constant c connects the two masses. a) Indicate qualitatively the form of the two normal modes for oscillations in the plane of the strings and spring. b) Write the equations of motion for small oscillations of the masses in terms of their horizontal displacements

x1 and x2. c) Find the normal mode frequencies and verify your answer to part (a) by finding the ratio x1/x2 for each mode.