Prerequisites

This course builds on the ideas introduced in NST Part IA “Physics”, using the tools of vector calculus taught in NST Part IA “Mathematics”.

Learning Outcomes and Assessment

The main areas covered are orbits, rigid body dynamics, normal modes and continuum mechanics (elasticity and fluids). However, the real purpose of this course, as with so many other courses in 1B Physics, is to demonstrate the enormous range of phenomena that are all explained by single set of fundamental laws or equations of Physics - in this case, Newton's Laws of Motion.

Synopsis

Newtonian mechanics, frames of reference: Review of Part IA mechanics: many-particle system, internal and external forces and energy. Central forces, motion in a plane. Non-inertial frames, rotating frames, centrifugal and Coriolis forces.

Orbits: Effective potential and radial motion, bound and unbound orbits. Inverse-square law orbits, circular and elliptic, Kepler's laws. Escape velocity, transfer orbits, gravitational slingshot. Hyperbolic orbits, angle of scattering, repulsive force. Two-body problem, reduced mass. General features of three-body problem. Brief treatment of tidal effects in gravitational systems.

Rigid body dynamics: Instantaneous motion of a rigid body, angular velocity and angular momentum, moment of inertia tensor, principal axes and moments. Rotational energy, inertia ellipsoid. Euler's equations, free precession of a symmetrical top, space and body frequencies. Forced precession, gyroscopes.

Introduction to Lagrangian mechanics: Generalised coordinates. Hamilton’s principle and Lagrange’s equations. Symmetries and conservation laws. Conservation of the Hamiltonian for time-independent systems.

Normal modes: Analysis of many-particle system in terms of normal modes. Degrees of freedom, matrix notation, zero-frequency and degenerate modes.  Motion in three dimensions, modes of molecules.

Elasticity: Hooke's law, Young's modulus, Poisson's ratio. Bulk modulus, shear modulus, stress tensor, principal stresses, strain tensor. Elastic energy. Torsion of cylinder. Bending of beams, bending moment, boundary conditions. Euler strut.  Brief treatment of elastic waves.

Fluid dynamics: Continuum fields, material derivatives, relation to particle paths and streamlines. Mass conservation, incompressibility. Convective derivative and equation of motion. Bernoulli's theorem, applications. Velocity potential, applications: sources and sinks; flow past a sphere and cylinder; vortices; Magnus effect. Viscosity, Couette and Poiseuille flow. Reynolds number, laminar and turbulent flow.

BOOKS

There are many books suitable for this course, including:

Fluid Dynamics for Physicists, Faber T E (Cambridge, 1995).

Lectures on Physics, Feynman R P, Leighton R B and Sands S L (Addison Wesley 1964).

Classical Mechanics, Kibble T W B and Berkshire F H (Imperial College 2004).