Applied Mathematics
MP236: Mechanics I (5 ECTS)
This is a mechanics course for students who have already been exposed to an elementary mechanics course. Topics covered include dimensional analysis, variational calculus, Lagrangian mechanics and rigid body motion.
Taught in Semester(s) I. Examined in Semester(s) I.
Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- re-write a law expressed in dimensional quantities in an equivalent dimensionless form using the Buckingham pi theorem;
- re-write a law expressed in dimensional quantities in an equivalent dimensionless form using the Buckingham pi theorem;
- use the concept of similarity in conjunction with dimensional analysis to aid in the design of scale models;
- solve some simple optimisation problems in the calculus of variations using the Euler Lagrange equations;
- obtain the equations of motion of mechanical systems using the Lagrangian formulation of mechanics;
- mathematically model the motion of a rigid body, and solve some simple problems involving rigid bodies.
Indicative Content
- Dimensional analysis: fundamental units, derived units, dimensionless quantities, the Buckingham pi theorem, analysing systems using dimensional analysis, similarity, scale models
- Calculus of variations: some examples of variational problems - shortest distance between two points, minimal surface area of revolution, Fermat's principle. Derivation of the Euler-Lagrange equation, some first integrals of the Euler-Lagrange equation, solution of some problems, the Euler-Lagrange equations for several functions
- The Lagrangian formulation of mechanics: coordinate systems, degrees of freedom, generalised coordinates, holonomic systems, constraint forces, the action integral and Hamilton's principle, derivation of the Lagrange equations of motion for a holonomic system, examples of solving mechanics problems using Lagrange's equations
- Rigid body motion: the motion of the centre of mass of a system of particles, angular momentum and torque, motion about the centre of mass of a rigid body, angular velocity, the moment of inertia tensor, kinetic energy of a rigid body, the solution of some problems for rigid bodies
Module Resources
Classical Mechanics, 5th Edition, T. Kibble & F. Berkshire, Imperial College Press
Classical Mechanics, R. Gregory, Cambridge University Press
Introduction to the Foundations of Applied Mathematics, M. Holmes, Springer
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