Mechanics of Rigid Bodies and Fluids

Description

Overview: • To convey the generalisation of the mechanics of single-particle systems to many-particle systems. • To convey the central ideas of a continuum description of material behaviour and to understand relevant constraints. • To ground students in the basic principles governing three-dimensional motions of rigid bodies. • To convey how the ideas of continuum theory are applied to static and inviscid fluids. Optional for: M, MS, MT, MP Pre-requisites: MM201, MM202, MM205 or equivalent. Incompatible (Overlapping) Classes: Syllabus: Motivation: Mechanics of many-particle systems: Concepts of mass, centre of mass, momentum, angular momentum, force, energy. Newton's Laws in many-particle systems. The two-body problem under gravity. Continuum Theory: The continuum concept, density, distributed forces. Rigid body mechanics: A rigid body as a constrained many-particle system (fixed relative position of particles). Density, mass, centre of mass, angular velocity, angular momentum of a rigid body. Conservation of angular momentum. Moment of a force. Rigid body statics in three dimensions. Planar motion of a rigid body. Fluid systems: Pressure. The hydrostatic equation, force on a submerged surface, Archimedes principle. Kinematics: Euler and Lagrange descriptions, material time derivative, streamlines, particle paths, vorticity and circulation. Derivatives of material integrals; Reynolds' transport theorem. Mass conservation, incompressibility. Boundary conditions. Inviscid fluids, Euler's equation, Bernoulli's theorem; examples of simple flows. Learning Outcomes: On completion of this module, the student should • understand the concepts of force, linear and angular momentum, energy and Newton’s Laws in many-particle systems; • solve problems involving three-dimensional rigid bodies in equilibrium; • understand the concepts of angular momentum and the moment of a force, and solve simple problems involving two-dimensional motions of three-dimensional bodies; • understand the notion of pressure, and find the force on a submerged object; • understand basic kinematics of fluids; and • understand the basic properties of an inviscid fluid and apply Bernoulli’s theorem to simple flow problems. Assessment: 3 hour degree examination in December. July/August summer examination.