Aims & Learning Objectives:
Aims: To introduce the theory of three-dimensional vectors, their algebraic and geometrical properties and their use in mathematical modelling. To introduce Newtonian Mechanics by considering a selection of problems involving the dynamics of particles.
The student should be familiar with the laws of vector algebra and vector calculus and should be able to use them in the solution of 3D algebraic and geometrical problems. The student should also be able to use vectors to describe and model physical problems involving kinematics. The student should be able to apply Newton's second law of motion to derive governing equations of motion for problems of particle dynamics, and should also be able to analyse or solve such equations.
Vectors: Vector equations of lines and planes. Differentiation of vectors with respect to a scalar variable. Curvature. Cartesian, polar and spherical co-ordinates. Vector identities. Dot and cross product, vector and scalar triple product and determinants from geometric viewpoint. Basic concepts of mass, length and time, particles, force. Basic forces of nature: structure of matter, microscopic and macroscopic forces. Units and dimensions: dimensional analysis and scaling. Kinematics: the description of particle motion in terms of vectors, velocity and acceleration in polar coordinates, angular velocity, relative velocity. Newton's Laws: Kepler's laws, momentum, Newton's laws of motion, Newton's law of gravitation. Newtonian Mechanics of Particles: projectiles in a resisting medium, constrained particle motion; solution of the governing differential equations for a variety of problems. Central Forces: motion under a central force.